Agenda
Section 3.4
Reminders
Written HW 2 due 10/3 or 10/5
WebHW due 10/6
Office hours Tues, Thurs1-2 pm (5852 East Hall)
MathLab office hourSun 7-8 pm (MathLab)
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
§3.4 Complex Eigenvalues
Objectives
Be able to find real solutions to x′ = Ax when A hascomplex eigenvalues
Be able to properly label the critical point wheneigenvalues are complex
Be able to sketch a phase portrait for systems withcomplex eigenvalues
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Example
Find the general solution to
x′ = Ax,
where
A =
[1 −15 −3
]and plot the phase portrait.
Recall that we’re looking of solutions of the form
x = veλt ,
where λ and v are the eigenvalues and eigenvectors of A,respectively. We begin by finding the eigenvalues.
det(A− λI ) = 0
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Example
Find the general solution to
x′ = Ax,
where
A =
[1 −15 −3
]and plot the phase portrait.
Recall that we’re looking of solutions of the form
x = veλt ,
where λ and v are the eigenvalues and eigenvectors of A,respectively. We begin by finding the eigenvalues.
det(A− λI ) = 0
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Example
Find the general solution to
x′ = Ax,
where
A =
[1 −15 −3
]and plot the phase portrait.
Recall that we’re looking of solutions of the form
x = veλt ,
where λ and v are the eigenvalues and eigenvectors of A,respectively. We begin by finding the eigenvalues.
det(A− λI ) = 0
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
det(A− λI ) = 0∣∣∣∣ 1− λ −15 −3− λ
∣∣∣∣ = 0.
−(1− λ)(3 + λ)− (−1)5 = 0
λ2 + 2λ + 2 = 0
λ =−2±
√4− 4 · 22
λ = −1± i
Recall that complex roots of polynomials with real coefficientscome in conjugate pairs. In addition, we showed that thecorresponding eigenvectors also come in conjugate pairs.Therefore, we just need to find one eigenvector and then weshould be able to write down the second one. Let’s start byfinding the eigenvector with λ = −1 + i .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
det(A− λI ) = 0∣∣∣∣ 1− λ −15 −3− λ
∣∣∣∣ = 0.
−(1− λ)(3 + λ)− (−1)5 = 0
λ2 + 2λ + 2 = 0
λ =−2±
√4− 4 · 22
λ = −1± i
Recall that complex roots of polynomials with real coefficientscome in conjugate pairs. In addition, we showed that thecorresponding eigenvectors also come in conjugate pairs.Therefore, we just need to find one eigenvector and then weshould be able to write down the second one. Let’s start byfinding the eigenvector with λ = −1 + i .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
det(A− λI ) = 0∣∣∣∣ 1− λ −15 −3− λ
∣∣∣∣ = 0.
−(1− λ)(3 + λ)− (−1)5 = 0
λ2 + 2λ + 2 = 0
λ =−2±
√4− 4 · 22
λ = −1± i
Recall that complex roots of polynomials with real coefficientscome in conjugate pairs. In addition, we showed that thecorresponding eigenvectors also come in conjugate pairs.Therefore, we just need to find one eigenvector and then weshould be able to write down the second one. Let’s start byfinding the eigenvector with λ = −1 + i .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
det(A− λI ) = 0∣∣∣∣ 1− λ −15 −3− λ
∣∣∣∣ = 0.
−(1− λ)(3 + λ)− (−1)5 = 0
λ2 + 2λ + 2 = 0
λ =−2±
√4− 4 · 22
λ = −1± i
Recall that complex roots of polynomials with real coefficientscome in conjugate pairs. In addition, we showed that thecorresponding eigenvectors also come in conjugate pairs.Therefore, we just need to find one eigenvector and then weshould be able to write down the second one. Let’s start byfinding the eigenvector with λ = −1 + i .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
det(A− λI ) = 0∣∣∣∣ 1− λ −15 −3− λ
∣∣∣∣ = 0.
−(1− λ)(3 + λ)− (−1)5 = 0
λ2 + 2λ + 2 = 0
λ =−2±
√4− 4 · 22
λ = −1± i
Recall that complex roots of polynomials with real coefficientscome in conjugate pairs. In addition, we showed that thecorresponding eigenvectors also come in conjugate pairs.Therefore, we just need to find one eigenvector and then weshould be able to write down the second one. Let’s start byfinding the eigenvector with λ = −1 + i .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
det(A− λI ) = 0∣∣∣∣ 1− λ −15 −3− λ
∣∣∣∣ = 0.
−(1− λ)(3 + λ)− (−1)5 = 0
λ2 + 2λ + 2 = 0
λ =−2±
√4− 4 · 22
λ = −1± i
Recall that complex roots of polynomials with real coefficientscome in conjugate pairs. In addition, we showed that thecorresponding eigenvectors also come in conjugate pairs.Therefore, we just need to find one eigenvector and then weshould be able to write down the second one. Let’s start byfinding the eigenvector with λ = −1 + i .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
det(A− λI ) = 0∣∣∣∣ 1− λ −15 −3− λ
∣∣∣∣ = 0.
−(1− λ)(3 + λ)− (−1)5 = 0
λ2 + 2λ + 2 = 0
λ =−2±
√4− 4 · 22
λ = −1± i
Recall that complex roots of polynomials with real coefficientscome in conjugate pairs. In addition, we showed that thecorresponding eigenvectors also come in conjugate pairs.Therefore, we just need to find one eigenvector and then weshould be able to write down the second one. Let’s start byfinding the eigenvector with λ = −1 + i .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
To find the eigenvector with eigenvalue λ = −1 + i , we needto solve
(A− (−1 + i)I )v−1+i = 0([1 −15 −3
]− (−1 + i)
[1 00 1
])v−1+i = 0[
1 + 1− i −15 −3 + 1− i
]v−1+i = 0[
2− i −15 −2− i
]v−1+i = 0
We can express the system as an augmented matrix[2− i −1 0
5 −2− i 0
]
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
To find the eigenvector with eigenvalue λ = −1 + i , we needto solve
(A− (−1 + i)I )v−1+i = 0([1 −15 −3
]− (−1 + i)
[1 00 1
])v−1+i = 0[
1 + 1− i −15 −3 + 1− i
]v−1+i = 0[
2− i −15 −2− i
]v−1+i = 0
We can express the system as an augmented matrix[2− i −1 0
5 −2− i 0
]
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
To find the eigenvector with eigenvalue λ = −1 + i , we needto solve
(A− (−1 + i)I )v−1+i = 0([1 −15 −3
]− (−1 + i)
[1 00 1
])v−1+i = 0[
1 + 1− i −15 −3 + 1− i
]v−1+i = 0[
2− i −15 −2− i
]v−1+i = 0
We can express the system as an augmented matrix[2− i −1 0
5 −2− i 0
]
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
To find the eigenvector with eigenvalue λ = −1 + i , we needto solve
(A− (−1 + i)I )v−1+i = 0([1 −15 −3
]− (−1 + i)
[1 00 1
])v−1+i = 0[
1 + 1− i −15 −3 + 1− i
]v−1+i = 0[
2− i −15 −2− i
]v−1+i = 0
We can express the system as an augmented matrix[2− i −1 0
5 −2− i 0
]
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
To find the eigenvector with eigenvalue λ = −1 + i , we needto solve
(A− (−1 + i)I )v−1+i = 0([1 −15 −3
]− (−1 + i)
[1 00 1
])v−1+i = 0[
1 + 1− i −15 −3 + 1− i
]v−1+i = 0[
2− i −15 −2− i
]v−1+i = 0
We can express the system as an augmented matrix[2− i −1 0
5 −2− i 0
]
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
[2− i −1 0
5 −2− i 0
]Add
−5
2− iR1 to R2[
2− i −1 00 0 0
]The matrix is now in row echelon form. In this case, the firstcolumn is a pivot column, so x1 is a basic variable and x2 is afree variable. Therefore, we should set x2 equal to a parameterand try to solve for x1.
x2 = c
We can use the first equation to solve for x1. That is,
(2− i)x1 − 1c = 0 =⇒ x1 =c
2− i.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
[2− i −1 0
5 −2− i 0
]Add
−5
2− iR1 to R2[
2− i −1 00 0 0
]The matrix is now in row echelon form. In this case, the firstcolumn is a pivot column, so x1 is a basic variable and x2 is afree variable. Therefore, we should set x2 equal to a parameterand try to solve for x1.
x2 = c
We can use the first equation to solve for x1. That is,
(2− i)x1 − 1c = 0 =⇒ x1 =c
2− i.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
[2− i −1 0
5 −2− i 0
]Add
−5
2− iR1 to R2[
2− i −1 00 0 0
]The matrix is now in row echelon form. In this case, the firstcolumn is a pivot column, so x1 is a basic variable and x2 is afree variable. Therefore, we should set x2 equal to a parameterand try to solve for x1.
x2 = c
We can use the first equation to solve for x1. That is,
(2− i)x1 − 1c = 0 =⇒ x1 =c
2− i.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
[2− i −1 0
5 −2− i 0
]Add
−5
2− iR1 to R2[
2− i −1 00 0 0
]The matrix is now in row echelon form. In this case, the firstcolumn is a pivot column, so x1 is a basic variable and x2 is afree variable. Therefore, we should set x2 equal to a parameterand try to solve for x1.
x2 = c
We can use the first equation to solve for x1. That is,
(2− i)x1 − 1c = 0 =⇒ x1 =c
2− i.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
[2− i −1 0
5 −2− i 0
]Add
−5
2− iR1 to R2[
2− i −1 00 0 0
]The matrix is now in row echelon form. In this case, the firstcolumn is a pivot column, so x1 is a basic variable and x2 is afree variable. Therefore, we should set x2 equal to a parameterand try to solve for x1.
x2 = c
We can use the first equation to solve for x1. That is,
(2− i)x1 − 1c = 0 =⇒ x1 =c
2− i.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
[2− i −1 0
5 −2− i 0
]Add
−5
2− iR1 to R2[
2− i −1 00 0 0
]The matrix is now in row echelon form. In this case, the firstcolumn is a pivot column, so x1 is a basic variable and x2 is afree variable. Therefore, we should set x2 equal to a parameterand try to solve for x1.
x2 = c
We can use the first equation to solve for x1. That is,
(2− i)x1 − 1c = 0 =⇒ x1 =c
2− i.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
[2− i −1 0
5 −2− i 0
]Add
−5
2− iR1 to R2[
2− i −1 00 0 0
]The matrix is now in row echelon form. In this case, the firstcolumn is a pivot column, so x1 is a basic variable and x2 is afree variable. Therefore, we should set x2 equal to a parameterand try to solve for x1.
x2 = c
We can use the first equation to solve for x1. That is,
(2− i)x1 − 1c = 0 =⇒ x1 =c
2− i.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
We can now write out the eigenvector.
v−1+i =
[x1
x2
]=
[c/(2− i)
c
]= c
[1/(2− i)
1
]If we drop the parameter and multiply by 2− i to remove thefraction, we get
v−1+i =
[1
2− i
].
Since eigenvectors come in conjugate pairs when A is real, wehave
v−1−i =
[1
2 + i
]as the eigenvector when λ = −1− i .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
We can now write out the eigenvector.
v−1+i =
[x1
x2
]=
[c/(2− i)
c
]= c
[1/(2− i)
1
]If we drop the parameter and multiply by 2− i to remove thefraction, we get
v−1+i =
[1
2− i
].
Since eigenvectors come in conjugate pairs when A is real, wehave
v−1−i =
[1
2 + i
]as the eigenvector when λ = −1− i .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
We can now write out the eigenvector.
v−1+i =
[x1
x2
]=
[c/(2− i)
c
]= c
[1/(2− i)
1
]If we drop the parameter and multiply by 2− i to remove thefraction, we get
v−1+i =
[1
2− i
].
Since eigenvectors come in conjugate pairs when A is real, wehave
v−1−i =
[1
2 + i
]as the eigenvector when λ = −1− i .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
v−1+i =
[1
2− i
]and v−1−i =
[1
2 + i
]We can now write down two solutions
x−1+i =
[1
2− i
]e(−1+i)t and x−1−i =
[1
2 + i
]e(−1−i)t .
While we could multiply each solution by a constant and addthem together to construct the general solution, the main issueis that our solutions have complex terms which are hard tomake sense of, especially if we’re trying to model a real worldproblem. However, if we could find two linearly independent,real solutions from these complex solutions, then we could usethem to write down the general solution with real terms.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
v−1+i =
[1
2− i
]and v−1−i =
[1
2 + i
]We can now write down two solutions
x−1+i =
[1
2− i
]e(−1+i)t and x−1−i =
[1
2 + i
]e(−1−i)t .
While we could multiply each solution by a constant and addthem together to construct the general solution, the main issueis that our solutions have complex terms which are hard tomake sense of, especially if we’re trying to model a real worldproblem. However, if we could find two linearly independent,real solutions from these complex solutions, then we could usethem to write down the general solution with real terms.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Constructing Real Solutions from Complex Solutions
Suppose x′ = Ax has a complex eigenvalue λ = α + iβ witheigenvector
vα+iβ = a + ib,
where α, β, a, and b are real. Since complex eigenvalues andeigenvectors come in conjugate pairs, we also know thatλ = α− iβ and
vα−iβ = a− ib
are eigenvalues and eigenvectors of A, respectively. Therefore,two solutions of x′ = Ax are
xα+iβ = (a + ib)e(α+iβ)t
andxα−iβ = (a− ib)e(α−iβ)t .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Constructing Real Solutions from Complex Solutions
Suppose x′ = Ax has a complex eigenvalue λ = α + iβ witheigenvector
vα+iβ = a + ib,
where α, β, a, and b are real. Since complex eigenvalues andeigenvectors come in conjugate pairs, we also know thatλ = α− iβ and
vα−iβ = a− ib
are eigenvalues and eigenvectors of A, respectively. Therefore,two solutions of x′ = Ax are
xα+iβ = (a + ib)e(α+iβ)t
andxα−iβ = (a− ib)e(α−iβ)t .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Constructing Real Solutions from Complex Solutions
Suppose x′ = Ax has a complex eigenvalue λ = α + iβ witheigenvector
vα+iβ = a + ib,
where α, β, a, and b are real. Since complex eigenvalues andeigenvectors come in conjugate pairs, we also know thatλ = α− iβ and
vα−iβ = a− ib
are eigenvalues and eigenvectors of A, respectively. Therefore,two solutions of x′ = Ax are
xα+iβ = (a + ib)e(α+iβ)t
andxα−iβ = (a− ib)e(α−iβ)t .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Recall Euler’s formula:
e iθ = cos θ + i sin θ.
(This identity can be derived by adding the Taylor series for cos θand i sin θ.) We can now use this to rewrite xα+iβ and xα−iβ.That is,
xα+iβ = (a + ib)e(α+iβ)t
= (a + ib)eαte iβt
= (a + ib)eαt(cosβt + i sinβt)
andxα−iβ = (a− ib)e(α−iβ)t
= (a− ib)eαte−iβt
= (a− ib)eαt(cos (−βt) + i sin (−βt))
= (a− ib)eαt(cosβt − i sinβt).
In the last step, we used the fact that cosine is even and sine isodd.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Recall Euler’s formula:
e iθ = cos θ + i sin θ.
(This identity can be derived by adding the Taylor series for cos θand i sin θ.) We can now use this to rewrite xα+iβ and xα−iβ.That is,
xα+iβ = (a + ib)e(α+iβ)t
= (a + ib)eαte iβt
= (a + ib)eαt(cosβt + i sinβt)
andxα−iβ = (a− ib)e(α−iβ)t
= (a− ib)eαte−iβt
= (a− ib)eαt(cos (−βt) + i sin (−βt))
= (a− ib)eαt(cosβt − i sinβt).
In the last step, we used the fact that cosine is even and sine isodd.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Recall Euler’s formula:
e iθ = cos θ + i sin θ.
(This identity can be derived by adding the Taylor series for cos θand i sin θ.) We can now use this to rewrite xα+iβ and xα−iβ.That is,
xα+iβ = (a + ib)e(α+iβ)t
= (a + ib)eαte iβt
= (a + ib)eαt(cosβt + i sinβt)
andxα−iβ = (a− ib)e(α−iβ)t
= (a− ib)eαte−iβt
= (a− ib)eαt(cos (−βt) + i sin (−βt))
= (a− ib)eαt(cosβt − i sinβt).
In the last step, we used the fact that cosine is even and sine isodd.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Recall Euler’s formula:
e iθ = cos θ + i sin θ.
(This identity can be derived by adding the Taylor series for cos θand i sin θ.) We can now use this to rewrite xα+iβ and xα−iβ.That is,
xα+iβ = (a + ib)e(α+iβ)t
= (a + ib)eαte iβt
= (a + ib)eαt(cosβt + i sinβt)
andxα−iβ = (a− ib)e(α−iβ)t
= (a− ib)eαte−iβt
= (a− ib)eαt(cos (−βt) + i sin (−βt))
= (a− ib)eαt(cosβt − i sinβt).
In the last step, we used the fact that cosine is even and sine isodd.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Recall Euler’s formula:
e iθ = cos θ + i sin θ.
(This identity can be derived by adding the Taylor series for cos θand i sin θ.) We can now use this to rewrite xα+iβ and xα−iβ.That is,
xα+iβ = (a + ib)e(α+iβ)t
= (a + ib)eαte iβt
= (a + ib)eαt(cosβt + i sinβt)
andxα−iβ = (a− ib)e(α−iβ)t
= (a− ib)eαte−iβt
= (a− ib)eαt(cos (−βt) + i sin (−βt))
= (a− ib)eαt(cosβt − i sinβt).
In the last step, we used the fact that cosine is even and sine isodd.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Recall Euler’s formula:
e iθ = cos θ + i sin θ.
(This identity can be derived by adding the Taylor series for cos θand i sin θ.) We can now use this to rewrite xα+iβ and xα−iβ.That is,
xα+iβ = (a + ib)e(α+iβ)t
= (a + ib)eαte iβt
= (a + ib)eαt(cosβt + i sinβt)
andxα−iβ = (a− ib)e(α−iβ)t
= (a− ib)eαte−iβt
= (a− ib)eαt(cos (−βt) + i sin (−βt))
= (a− ib)eαt(cosβt − i sinβt).
In the last step, we used the fact that cosine is even and sine isodd.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Recall Euler’s formula:
e iθ = cos θ + i sin θ.
(This identity can be derived by adding the Taylor series for cos θand i sin θ.) We can now use this to rewrite xα+iβ and xα−iβ.That is,
xα+iβ = (a + ib)e(α+iβ)t
= (a + ib)eαte iβt
= (a + ib)eαt(cosβt + i sinβt)
andxα−iβ = (a− ib)e(α−iβ)t
= (a− ib)eαte−iβt
= (a− ib)eαt(cos (−βt) + i sin (−βt))
= (a− ib)eαt(cosβt − i sinβt).
In the last step, we used the fact that cosine is even and sine isodd.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Recall Euler’s formula:
e iθ = cos θ + i sin θ.
(This identity can be derived by adding the Taylor series for cos θand i sin θ.) We can now use this to rewrite xα+iβ and xα−iβ.That is,
xα+iβ = (a + ib)e(α+iβ)t
= (a + ib)eαte iβt
= (a + ib)eαt(cosβt + i sinβt)
andxα−iβ = (a− ib)e(α−iβ)t
= (a− ib)eαte−iβt
= (a− ib)eαt(cos (−βt) + i sin (−βt))
= (a− ib)eαt(cosβt − i sinβt).
In the last step, we used the fact that cosine is even and sine isodd.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
xα+iβ = (a + ib)eαt(cos βt + i sin βt)
xα−iβ = (a− ib)eαt(cos βt − i sin βt)
If we separate xα+iβ and xα−iβ into their real and imagineparts, we get
xα+iβ = eαt (a cos βt − b sin βt) + ieαt (a sin βt + b cos βt)
xα−iβ = eαt (a cos βt − b sin βt)− ieαt (a sin βt + b cos βt) .
By the principle of superposition, we know that xα+iβ + xα−iβis a solution to x′ = Ax. Computing this gives us
xα+iβ + xα−iβ = 2eαt (a cos βt − b sin βt) .
This is a real solution! For simplicity, let’s divide by 2. Ourfirst real solution is then
x1 = eαt (a cos βt − b sin βt) .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
xα+iβ = (a + ib)eαt(cos βt + i sin βt)
xα−iβ = (a− ib)eαt(cos βt − i sin βt)
If we separate xα+iβ and xα−iβ into their real and imagineparts, we get
xα+iβ = eαt (a cos βt − b sin βt) + ieαt (a sin βt + b cos βt)
xα−iβ = eαt (a cos βt − b sin βt)− ieαt (a sin βt + b cos βt) .
By the principle of superposition, we know that xα+iβ + xα−iβis a solution to x′ = Ax. Computing this gives us
xα+iβ + xα−iβ = 2eαt (a cos βt − b sin βt) .
This is a real solution! For simplicity, let’s divide by 2. Ourfirst real solution is then
x1 = eαt (a cos βt − b sin βt) .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
xα+iβ = (a + ib)eαt(cos βt + i sin βt)
xα−iβ = (a− ib)eαt(cos βt − i sin βt)
If we separate xα+iβ and xα−iβ into their real and imagineparts, we get
xα+iβ = eαt (a cos βt − b sin βt) + ieαt (a sin βt + b cos βt)
xα−iβ = eαt (a cos βt − b sin βt)− ieαt (a sin βt + b cos βt) .
By the principle of superposition, we know that xα+iβ + xα−iβis a solution to x′ = Ax. Computing this gives us
xα+iβ + xα−iβ = 2eαt (a cos βt − b sin βt) .
This is a real solution! For simplicity, let’s divide by 2. Ourfirst real solution is then
x1 = eαt (a cos βt − b sin βt) .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
xα+iβ = (a + ib)eαt(cos βt + i sin βt)
xα−iβ = (a− ib)eαt(cos βt − i sin βt)
If we separate xα+iβ and xα−iβ into their real and imagineparts, we get
xα+iβ = eαt (a cos βt − b sin βt) + ieαt (a sin βt + b cos βt)
xα−iβ = eαt (a cos βt − b sin βt)− ieαt (a sin βt + b cos βt) .
By the principle of superposition, we know that xα+iβ + xα−iβis a solution to x′ = Ax. Computing this gives us
xα+iβ + xα−iβ = 2eαt (a cos βt − b sin βt) .
This is a real solution! For simplicity, let’s divide by 2. Ourfirst real solution is then
x1 = eαt (a cos βt − b sin βt) .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
To find our second real solution, let’s compute xα+iβ − xα−iβ.That is,
xα+iβ − xα−iβ = 2ieαt (a sin βt + b cos βt) .
Recall that if we divide a solution by a nonzero scalar, it’s stilla solution, even if the scalar is complex. Therefore, if wedivide xα+iβ − xα−iβ by 2i , we’ll have our second real solution.That is,
x2 = eαt (a sin βt + b cos βt) .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
To find our second real solution, let’s compute xα+iβ − xα−iβ.That is,
xα+iβ − xα−iβ = 2ieαt (a sin βt + b cos βt) .
Recall that if we divide a solution by a nonzero scalar, it’s stilla solution, even if the scalar is complex. Therefore, if wedivide xα+iβ − xα−iβ by 2i , we’ll have our second real solution.That is,
x2 = eαt (a sin βt + b cos βt) .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
To summarize, if you happen to get a complex eigenvalue, start bywriting the solution in the form veλt , like we normally would:
xα+iβ = (a + ib)e(α+iβ)t .
Next, use Euler’s formula, e iθ = cos θ + i sin θ, to rewrite theexponential in terms of sine and cosine. That is,
xα+iβ = (a + ib)eαt(cosβt + i sinβt).
After that, you should separate the real and imaginary terms to get
xα+iβ = eαt (a cosβt − b sinβt) + ieαt (a sinβt + b cosβt) .
We’ve shown that Re(xα+iβ) and Im(xα+iβ) both satisfy x′ = Ax.Therefore, we can use the principle of superposition to write oursolution as
x = c1eαt (a cosβt − b sinβt) + c2e
αt (a sinβt + b cosβt) .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
To summarize, if you happen to get a complex eigenvalue, start bywriting the solution in the form veλt , like we normally would:
xα+iβ = (a + ib)e(α+iβ)t .
Next, use Euler’s formula, e iθ = cos θ + i sin θ, to rewrite theexponential in terms of sine and cosine. That is,
xα+iβ = (a + ib)eαt(cosβt + i sinβt).
After that, you should separate the real and imaginary terms to get
xα+iβ = eαt (a cosβt − b sinβt) + ieαt (a sinβt + b cosβt) .
We’ve shown that Re(xα+iβ) and Im(xα+iβ) both satisfy x′ = Ax.Therefore, we can use the principle of superposition to write oursolution as
x = c1eαt (a cosβt − b sinβt) + c2e
αt (a sinβt + b cosβt) .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
To summarize, if you happen to get a complex eigenvalue, start bywriting the solution in the form veλt , like we normally would:
xα+iβ = (a + ib)e(α+iβ)t .
Next, use Euler’s formula, e iθ = cos θ + i sin θ, to rewrite theexponential in terms of sine and cosine. That is,
xα+iβ = (a + ib)eαt(cosβt + i sinβt).
After that, you should separate the real and imaginary terms to get
xα+iβ = eαt (a cosβt − b sinβt) + ieαt (a sinβt + b cosβt) .
We’ve shown that Re(xα+iβ) and Im(xα+iβ) both satisfy x′ = Ax.Therefore, we can use the principle of superposition to write oursolution as
x = c1eαt (a cosβt − b sinβt) + c2e
αt (a sinβt + b cosβt) .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
To summarize, if you happen to get a complex eigenvalue, start bywriting the solution in the form veλt , like we normally would:
xα+iβ = (a + ib)e(α+iβ)t .
Next, use Euler’s formula, e iθ = cos θ + i sin θ, to rewrite theexponential in terms of sine and cosine. That is,
xα+iβ = (a + ib)eαt(cosβt + i sinβt).
After that, you should separate the real and imaginary terms to get
xα+iβ = eαt (a cosβt − b sinβt) + ieαt (a sinβt + b cosβt) .
We’ve shown that Re(xα+iβ) and Im(xα+iβ) both satisfy x′ = Ax.Therefore, we can use the principle of superposition to write oursolution as
x = c1eαt (a cosβt − b sinβt) + c2e
αt (a sinβt + b cosβt) .
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Let’s use our solution from the example to find two real solutions to x′ = Ax. Recallthat our first solution was
x−1+i =
[1
2− i
]e(−1+i)t .
We need to first rewrite the exponential term using Euler’s formula.
x−1+i =
[1
2− i
]e(−1+i)t
=
[1
2− i
]e−te it
=
[1
2− i
]e−t (cos t + i sin t)
Next, we need to separate real and imaginary terms.
x−1+i =
[1
2− i
]e−t (cos t + i sin t)
=
([12
]+ i
[0−1
])e−t (cos t + i sin t)
= e−t
([12
]cos t −
[0−1
]sin t
)+ ie−t
([12
]sin t +
[0−1
]cos t
)
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Let’s use our solution from the example to find two real solutions to x′ = Ax. Recallthat our first solution was
x−1+i =
[1
2− i
]e(−1+i)t .
We need to first rewrite the exponential term using Euler’s formula.
x−1+i =
[1
2− i
]e(−1+i)t
=
[1
2− i
]e−te it
=
[1
2− i
]e−t (cos t + i sin t)
Next, we need to separate real and imaginary terms.
x−1+i =
[1
2− i
]e−t (cos t + i sin t)
=
([12
]+ i
[0−1
])e−t (cos t + i sin t)
= e−t
([12
]cos t −
[0−1
]sin t
)+ ie−t
([12
]sin t +
[0−1
]cos t
)
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Let’s use our solution from the example to find two real solutions to x′ = Ax. Recallthat our first solution was
x−1+i =
[1
2− i
]e(−1+i)t .
We need to first rewrite the exponential term using Euler’s formula.
x−1+i =
[1
2− i
]e(−1+i)t
=
[1
2− i
]e−te it
=
[1
2− i
]e−t (cos t + i sin t)
Next, we need to separate real and imaginary terms.
x−1+i =
[1
2− i
]e−t (cos t + i sin t)
=
([12
]+ i
[0−1
])e−t (cos t + i sin t)
= e−t
([12
]cos t −
[0−1
]sin t
)+ ie−t
([12
]sin t +
[0−1
]cos t
)
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Let’s use our solution from the example to find two real solutions to x′ = Ax. Recallthat our first solution was
x−1+i =
[1
2− i
]e(−1+i)t .
We need to first rewrite the exponential term using Euler’s formula.
x−1+i =
[1
2− i
]e(−1+i)t
=
[1
2− i
]e−te it
=
[1
2− i
]e−t (cos t + i sin t)
Next, we need to separate real and imaginary terms.
x−1+i =
[1
2− i
]e−t (cos t + i sin t)
=
([12
]+ i
[0−1
])e−t (cos t + i sin t)
= e−t
([12
]cos t −
[0−1
]sin t
)+ ie−t
([12
]sin t +
[0−1
]cos t
)
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Let’s use our solution from the example to find two real solutions to x′ = Ax. Recallthat our first solution was
x−1+i =
[1
2− i
]e(−1+i)t .
We need to first rewrite the exponential term using Euler’s formula.
x−1+i =
[1
2− i
]e(−1+i)t
=
[1
2− i
]e−te it
=
[1
2− i
]e−t (cos t + i sin t)
Next, we need to separate real and imaginary terms.
x−1+i =
[1
2− i
]e−t (cos t + i sin t)
=
([12
]+ i
[0−1
])e−t (cos t + i sin t)
= e−t
([12
]cos t −
[0−1
]sin t
)+ ie−t
([12
]sin t +
[0−1
]cos t
)
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Let’s use our solution from the example to find two real solutions to x′ = Ax. Recallthat our first solution was
x−1+i =
[1
2− i
]e(−1+i)t .
We need to first rewrite the exponential term using Euler’s formula.
x−1+i =
[1
2− i
]e(−1+i)t
=
[1
2− i
]e−te it
=
[1
2− i
]e−t (cos t + i sin t)
Next, we need to separate real and imaginary terms.
x−1+i =
[1
2− i
]e−t (cos t + i sin t)
=
([12
]+ i
[0−1
])e−t (cos t + i sin t)
= e−t
([12
]cos t −
[0−1
]sin t
)+ ie−t
([12
]sin t +
[0−1
]cos t
)
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Let’s use our solution from the example to find two real solutions to x′ = Ax. Recallthat our first solution was
x−1+i =
[1
2− i
]e(−1+i)t .
We need to first rewrite the exponential term using Euler’s formula.
x−1+i =
[1
2− i
]e(−1+i)t
=
[1
2− i
]e−te it
=
[1
2− i
]e−t (cos t + i sin t)
Next, we need to separate real and imaginary terms.
x−1+i =
[1
2− i
]e−t (cos t + i sin t)
=
([12
]+ i
[0−1
])e−t (cos t + i sin t)
= e−t
([12
]cos t −
[0−1
]sin t
)+ ie−t
([12
]sin t +
[0−1
]cos t
)
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
x−1+i = e−t
([12
]cos t −
[0−1
]sin t
)+ ie−t
([12
]sin t +
[0−1
]cos t
)
Since we know that Re(x−1+i ) and Im(x−1+i ) are solutions to x′ = Ax we can use theprinciple of superposition to write the solution as
x = c1e−t
([12
]cos t −
[0−1
]sin t
)+ c2e
−t
([12
]sin t +
[0−1
]cos t
)
or more compactly as
x = c1e−t
[cos t
2 cos t + sin t
]+ c2e
−t
[sin t
2 sin t − cos t
].
To check that our solutions form a fundamental set, we can compute the Wronskianand show it’s nonzero.
W [x1, x2](t) =
∣∣∣∣ e−t cos t e−t sin te−t(2 cos t + sin t) e−t(2 sin t − cos t)
∣∣∣∣(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
x−1+i = e−t
([12
]cos t −
[0−1
]sin t
)+ ie−t
([12
]sin t +
[0−1
]cos t
)
Since we know that Re(x−1+i ) and Im(x−1+i ) are solutions to x′ = Ax we can use theprinciple of superposition to write the solution as
x = c1e−t
([12
]cos t −
[0−1
]sin t
)+ c2e
−t
([12
]sin t +
[0−1
]cos t
)
or more compactly as
x = c1e−t
[cos t
2 cos t + sin t
]+ c2e
−t
[sin t
2 sin t − cos t
].
To check that our solutions form a fundamental set, we can compute the Wronskianand show it’s nonzero.
W [x1, x2](t) =
∣∣∣∣ e−t cos t e−t sin te−t(2 cos t + sin t) e−t(2 sin t − cos t)
∣∣∣∣(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
x−1+i = e−t
([12
]cos t −
[0−1
]sin t
)+ ie−t
([12
]sin t +
[0−1
]cos t
)
Since we know that Re(x−1+i ) and Im(x−1+i ) are solutions to x′ = Ax we can use theprinciple of superposition to write the solution as
x = c1e−t
([12
]cos t −
[0−1
]sin t
)+ c2e
−t
([12
]sin t +
[0−1
]cos t
)
or more compactly as
x = c1e−t
[cos t
2 cos t + sin t
]+ c2e
−t
[sin t
2 sin t − cos t
].
To check that our solutions form a fundamental set, we can compute the Wronskianand show it’s nonzero.
W [x1, x2](t) =
∣∣∣∣ e−t cos t e−t sin te−t(2 cos t + sin t) e−t(2 sin t − cos t)
∣∣∣∣(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
x−1+i = e−t
([12
]cos t −
[0−1
]sin t
)+ ie−t
([12
]sin t +
[0−1
]cos t
)
Since we know that Re(x−1+i ) and Im(x−1+i ) are solutions to x′ = Ax we can use theprinciple of superposition to write the solution as
x = c1e−t
([12
]cos t −
[0−1
]sin t
)+ c2e
−t
([12
]sin t +
[0−1
]cos t
)
or more compactly as
x = c1e−t
[cos t
2 cos t + sin t
]+ c2e
−t
[sin t
2 sin t − cos t
].
To check that our solutions form a fundamental set, we can compute the Wronskianand show it’s nonzero.
W [x1, x2](t) =
∣∣∣∣ e−t cos t e−t sin te−t(2 cos t + sin t) e−t(2 sin t − cos t)
∣∣∣∣(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
W [x1, x2](t) =
∣∣∣∣ e−t cos t e−t sin te−t(2 cos t + sin t) e−t(2 sin t − cos t)
∣∣∣∣= e−2t(cos t)(2 sin t − cos t)− e−2t(sin t)(2 cos t + sin t)
= e−2t(2 sin t cos t − cos2 t)− e−2t(2 sin t cos t + sin2 t)
= e−2t(2 sin t cos t − cos2 t − 2 sin t cos t − sin2 t)
= −e−2t(cos2 t + sin2 t)
= −e−2t
Since −e−2t 6= 0 for all t, we can conclude that our solution is thegeneral solution.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
W [x1, x2](t) =
∣∣∣∣ e−t cos t e−t sin te−t(2 cos t + sin t) e−t(2 sin t − cos t)
∣∣∣∣= e−2t(cos t)(2 sin t − cos t)− e−2t(sin t)(2 cos t + sin t)
= e−2t(2 sin t cos t − cos2 t)− e−2t(2 sin t cos t + sin2 t)
= e−2t(2 sin t cos t − cos2 t − 2 sin t cos t − sin2 t)
= −e−2t(cos2 t + sin2 t)
= −e−2t
Since −e−2t 6= 0 for all t, we can conclude that our solution is thegeneral solution.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
W [x1, x2](t) =
∣∣∣∣ e−t cos t e−t sin te−t(2 cos t + sin t) e−t(2 sin t − cos t)
∣∣∣∣= e−2t(cos t)(2 sin t − cos t)− e−2t(sin t)(2 cos t + sin t)
= e−2t(2 sin t cos t − cos2 t)− e−2t(2 sin t cos t + sin2 t)
= e−2t(2 sin t cos t − cos2 t − 2 sin t cos t − sin2 t)
= −e−2t(cos2 t + sin2 t)
= −e−2t
Since −e−2t 6= 0 for all t, we can conclude that our solution is thegeneral solution.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
W [x1, x2](t) =
∣∣∣∣ e−t cos t e−t sin te−t(2 cos t + sin t) e−t(2 sin t − cos t)
∣∣∣∣= e−2t(cos t)(2 sin t − cos t)− e−2t(sin t)(2 cos t + sin t)
= e−2t(2 sin t cos t − cos2 t)− e−2t(2 sin t cos t + sin2 t)
= e−2t(2 sin t cos t − cos2 t − 2 sin t cos t − sin2 t)
= −e−2t(cos2 t + sin2 t)
= −e−2t
Since −e−2t 6= 0 for all t, we can conclude that our solution is thegeneral solution.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
W [x1, x2](t) =
∣∣∣∣ e−t cos t e−t sin te−t(2 cos t + sin t) e−t(2 sin t − cos t)
∣∣∣∣= e−2t(cos t)(2 sin t − cos t)− e−2t(sin t)(2 cos t + sin t)
= e−2t(2 sin t cos t − cos2 t)− e−2t(2 sin t cos t + sin2 t)
= e−2t(2 sin t cos t − cos2 t − 2 sin t cos t − sin2 t)
= −e−2t(cos2 t + sin2 t)
= −e−2t
Since −e−2t 6= 0 for all t, we can conclude that our solution is thegeneral solution.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
W [x1, x2](t) =
∣∣∣∣ e−t cos t e−t sin te−t(2 cos t + sin t) e−t(2 sin t − cos t)
∣∣∣∣= e−2t(cos t)(2 sin t − cos t)− e−2t(sin t)(2 cos t + sin t)
= e−2t(2 sin t cos t − cos2 t)− e−2t(2 sin t cos t + sin2 t)
= e−2t(2 sin t cos t − cos2 t − 2 sin t cos t − sin2 t)
= −e−2t(cos2 t + sin2 t)
= −e−2t
Since −e−2t 6= 0 for all t, we can conclude that our solution is thegeneral solution.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
W [x1, x2](t) =
∣∣∣∣ e−t cos t e−t sin te−t(2 cos t + sin t) e−t(2 sin t − cos t)
∣∣∣∣= e−2t(cos t)(2 sin t − cos t)− e−2t(sin t)(2 cos t + sin t)
= e−2t(2 sin t cos t − cos2 t)− e−2t(2 sin t cos t + sin2 t)
= e−2t(2 sin t cos t − cos2 t − 2 sin t cos t − sin2 t)
= −e−2t(cos2 t + sin2 t)
= −e−2t
Since −e−2t 6= 0 for all t, we can conclude that our solution is thegeneral solution.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
x = c1e−t
[cos t
2 cos t + sin t
]+ c2e
−t[
sin t2 sin t − cos t
]Recall that our eigenvalues were λ = −1± i . The imaginary part of theeigenvalue leads to rotations in our phase portrait due to the sines andcosines in our solution. In addition, the negative real part causestrajectories to tend to the critical point at the origin. As a result we endup getting a spiral sink which is asymptotically stable. To determinewhich direction the spiral is spinning, just evaluate x′ = Ax at a point.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Theorem
Let A have real or complex eigenvalues λ1 and λ2 such thatλ1 6= λ2, and let the corresponding eigenvectors be
v1 =
[x11
x21
]and v2 =
[x12
x22
].
Then
det
([x11 x12
x21 x22
])6= 0.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Proof
Suppose that
det
([x11 x12
x21 x22
])= 0. (∗)
We will show that this leads to a contradiction. If (∗) weretrue, then the system[
x11 x12
x21 x22
] [c1
c2
]= 0 (?)
would have a nonzero solution. However, (?) can be rewrittenas
c1
[x11
x21
]+ c2
[x12
x22
]= 0,
or simplyc1v1 + c2v2 = 0.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
c1v1 + c2v2 = 0 (#)
Since there exists a nonzero solution to (#), v1 and v2 must belinearly dependent. Therefore, there exists a constant k such that
v1 = kv2.
Furthermore, k 6= 0 since eigenvectors by definition are nonzero. Ifwe multiply both sides by A, we get that
Av1 = kAv2
λ1v1 = kλ2v2.
If we plug in v1 = kv2 for v1, we see that
λ1kv2 = kλ2v2
k(λ1 − λ2)v2 = 0.
Since k 6= 0 and v2 6= 0 by definition, we must have that λ1 = λ2.However, this contradicts that assumption that λ1 6= λ2.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Example
Suppose that a matrix A has complex eigenvalues λ = α± iβ,where β 6= 0, and eigenvectors v = a± ib. Show that
x1 = eαt (a cosβt − b sinβt)
andx2 = eαt (a sinβt + b cosβt)
form a fundamental set of solutions to x′ = Ax.
We can show that x1 and x2 form a fundamental set of solutions ifwe can show that
W [x1, x2](t) 6= 0, for all t.
We start by computing the Wronskian.
W [x1, x2](t) =
∣∣∣∣ eαt(a1 cosβt − b1 sinβt) eαt(a1 sinβt + b1 cosβt)eαt(a2 cosβt − b2 sinβt) eαt(a2 sinβt + b2 cosβt)
∣∣∣∣(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Example
Suppose that a matrix A has complex eigenvalues λ = α± iβ,where β 6= 0, and eigenvectors v = a± ib. Show that
x1 = eαt (a cosβt − b sinβt)
andx2 = eαt (a sinβt + b cosβt)
form a fundamental set of solutions to x′ = Ax.
We can show that x1 and x2 form a fundamental set of solutions ifwe can show that
W [x1, x2](t) 6= 0, for all t.
We start by computing the Wronskian.
W [x1, x2](t) =
∣∣∣∣ eαt(a1 cosβt − b1 sinβt) eαt(a1 sinβt + b1 cosβt)eαt(a2 cosβt − b2 sinβt) eαt(a2 sinβt + b2 cosβt)
∣∣∣∣(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Example
Suppose that a matrix A has complex eigenvalues λ = α± iβ,where β 6= 0, and eigenvectors v = a± ib. Show that
x1 = eαt (a cosβt − b sinβt)
andx2 = eαt (a sinβt + b cosβt)
form a fundamental set of solutions to x′ = Ax.
We can show that x1 and x2 form a fundamental set of solutions ifwe can show that
W [x1, x2](t) 6= 0, for all t.
We start by computing the Wronskian.
W [x1, x2](t) =
∣∣∣∣ eαt(a1 cosβt − b1 sinβt) eαt(a1 sinβt + b1 cosβt)eαt(a2 cosβt − b2 sinβt) eαt(a2 sinβt + b2 cosβt)
∣∣∣∣(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
W [x1, x2](t) =
∣∣∣∣ eαt(a1 cosβt − b1 sinβt) eαt(a1 sinβt + b1 cosβt)eαt(a2 cosβt − b2 sinβt) eαt(a2 sinβt + b2 cosβt)
∣∣∣∣=e2αt(a1 cosβt − b1 sinβt)(a2 sinβt + b2 cosβt)
−e2αt(a1 sinβt + b1 cosβt)(a2 cosβt − b2 sinβt)
=e2αt[a1b2(cos2 βt + sin2 βt)− a2b1(sin2 βt + cos2 βt)
]=e2αt(a1b2 − a2b1)
We know that e2αt 6= 0 for all t. Now we just need to show thata1b2 − a2b1 6= 0. Recall that
vα+iβ =
[a1
a2
]+ i
[b1
b2
]and vα−iβ =
[a1
a2
]− i
[b1
b2
].
Since ∣∣∣∣ a1 + ib1 a1 − ib1
a2 + ib2 a2 − ib2
∣∣∣∣ = −2i(a1b2 − a2b1) 6= 0
by the previous theorem, we can conclude that a1b2 − a2b1 6= 0 andhence W [x1, x2](t) 6= 0. Therefore, x1 and x2 form a fundamental set ofsolutions.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
W [x1, x2](t) =
∣∣∣∣ eαt(a1 cosβt − b1 sinβt) eαt(a1 sinβt + b1 cosβt)eαt(a2 cosβt − b2 sinβt) eαt(a2 sinβt + b2 cosβt)
∣∣∣∣=e2αt(a1 cosβt − b1 sinβt)(a2 sinβt + b2 cosβt)
−e2αt(a1 sinβt + b1 cosβt)(a2 cosβt − b2 sinβt)
=e2αt[a1b2(cos2 βt + sin2 βt)− a2b1(sin2 βt + cos2 βt)
]=e2αt(a1b2 − a2b1)
We know that e2αt 6= 0 for all t. Now we just need to show thata1b2 − a2b1 6= 0. Recall that
vα+iβ =
[a1
a2
]+ i
[b1
b2
]and vα−iβ =
[a1
a2
]− i
[b1
b2
].
Since ∣∣∣∣ a1 + ib1 a1 − ib1
a2 + ib2 a2 − ib2
∣∣∣∣ = −2i(a1b2 − a2b1) 6= 0
by the previous theorem, we can conclude that a1b2 − a2b1 6= 0 andhence W [x1, x2](t) 6= 0. Therefore, x1 and x2 form a fundamental set ofsolutions.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
W [x1, x2](t) =
∣∣∣∣ eαt(a1 cosβt − b1 sinβt) eαt(a1 sinβt + b1 cosβt)eαt(a2 cosβt − b2 sinβt) eαt(a2 sinβt + b2 cosβt)
∣∣∣∣=e2αt(a1 cosβt − b1 sinβt)(a2 sinβt + b2 cosβt)
−e2αt(a1 sinβt + b1 cosβt)(a2 cosβt − b2 sinβt)
=e2αt[a1b2(cos2 βt + sin2 βt)− a2b1(sin2 βt + cos2 βt)
]=e2αt(a1b2 − a2b1)
We know that e2αt 6= 0 for all t. Now we just need to show thata1b2 − a2b1 6= 0. Recall that
vα+iβ =
[a1
a2
]+ i
[b1
b2
]and vα−iβ =
[a1
a2
]− i
[b1
b2
].
Since ∣∣∣∣ a1 + ib1 a1 − ib1
a2 + ib2 a2 − ib2
∣∣∣∣ = −2i(a1b2 − a2b1) 6= 0
by the previous theorem, we can conclude that a1b2 − a2b1 6= 0 andhence W [x1, x2](t) 6= 0. Therefore, x1 and x2 form a fundamental set ofsolutions.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
W [x1, x2](t) =
∣∣∣∣ eαt(a1 cosβt − b1 sinβt) eαt(a1 sinβt + b1 cosβt)eαt(a2 cosβt − b2 sinβt) eαt(a2 sinβt + b2 cosβt)
∣∣∣∣=e2αt(a1 cosβt − b1 sinβt)(a2 sinβt + b2 cosβt)
−e2αt(a1 sinβt + b1 cosβt)(a2 cosβt − b2 sinβt)
=e2αt[a1b2(cos2 βt + sin2 βt)− a2b1(sin2 βt + cos2 βt)
]=e2αt(a1b2 − a2b1)
We know that e2αt 6= 0 for all t. Now we just need to show thata1b2 − a2b1 6= 0. Recall that
vα+iβ =
[a1
a2
]+ i
[b1
b2
]and vα−iβ =
[a1
a2
]− i
[b1
b2
].
Since ∣∣∣∣ a1 + ib1 a1 − ib1
a2 + ib2 a2 − ib2
∣∣∣∣ = −2i(a1b2 − a2b1) 6= 0
by the previous theorem, we can conclude that a1b2 − a2b1 6= 0 andhence W [x1, x2](t) 6= 0. Therefore, x1 and x2 form a fundamental set ofsolutions.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
W [x1, x2](t) =
∣∣∣∣ eαt(a1 cosβt − b1 sinβt) eαt(a1 sinβt + b1 cosβt)eαt(a2 cosβt − b2 sinβt) eαt(a2 sinβt + b2 cosβt)
∣∣∣∣=e2αt(a1 cosβt − b1 sinβt)(a2 sinβt + b2 cosβt)
−e2αt(a1 sinβt + b1 cosβt)(a2 cosβt − b2 sinβt)
=e2αt[a1b2(cos2 βt + sin2 βt)− a2b1(sin2 βt + cos2 βt)
]=e2αt(a1b2 − a2b1)
We know that e2αt 6= 0 for all t. Now we just need to show thata1b2 − a2b1 6= 0. Recall that
vα+iβ =
[a1
a2
]+ i
[b1
b2
]and vα−iβ =
[a1
a2
]− i
[b1
b2
].
Since ∣∣∣∣ a1 + ib1 a1 − ib1
a2 + ib2 a2 − ib2
∣∣∣∣ = −2i(a1b2 − a2b1) 6= 0
by the previous theorem, we can conclude that a1b2 − a2b1 6= 0 andhence W [x1, x2](t) 6= 0. Therefore, x1 and x2 form a fundamental set ofsolutions.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
W [x1, x2](t) =
∣∣∣∣ eαt(a1 cosβt − b1 sinβt) eαt(a1 sinβt + b1 cosβt)eαt(a2 cosβt − b2 sinβt) eαt(a2 sinβt + b2 cosβt)
∣∣∣∣=e2αt(a1 cosβt − b1 sinβt)(a2 sinβt + b2 cosβt)
−e2αt(a1 sinβt + b1 cosβt)(a2 cosβt − b2 sinβt)
=e2αt[a1b2(cos2 βt + sin2 βt)− a2b1(sin2 βt + cos2 βt)
]=e2αt(a1b2 − a2b1)
We know that e2αt 6= 0 for all t. Now we just need to show thata1b2 − a2b1 6= 0. Recall that
vα+iβ =
[a1
a2
]+ i
[b1
b2
]and vα−iβ =
[a1
a2
]− i
[b1
b2
].
Since ∣∣∣∣ a1 + ib1 a1 − ib1
a2 + ib2 a2 − ib2
∣∣∣∣ = −2i(a1b2 − a2b1) 6= 0
by the previous theorem, we can conclude that a1b2 − a2b1 6= 0 andhence W [x1, x2](t) 6= 0. Therefore, x1 and x2 form a fundamental set ofsolutions.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
W [x1, x2](t) =
∣∣∣∣ eαt(a1 cosβt − b1 sinβt) eαt(a1 sinβt + b1 cosβt)eαt(a2 cosβt − b2 sinβt) eαt(a2 sinβt + b2 cosβt)
∣∣∣∣=e2αt(a1 cosβt − b1 sinβt)(a2 sinβt + b2 cosβt)
−e2αt(a1 sinβt + b1 cosβt)(a2 cosβt − b2 sinβt)
=e2αt[a1b2(cos2 βt + sin2 βt)− a2b1(sin2 βt + cos2 βt)
]=e2αt(a1b2 − a2b1)
We know that e2αt 6= 0 for all t. Now we just need to show thata1b2 − a2b1 6= 0. Recall that
vα+iβ =
[a1
a2
]+ i
[b1
b2
]and vα−iβ =
[a1
a2
]− i
[b1
b2
].
Since ∣∣∣∣ a1 + ib1 a1 − ib1
a2 + ib2 a2 − ib2
∣∣∣∣ = −2i(a1b2 − a2b1) 6= 0
by the previous theorem, we can conclude that a1b2 − a2b1 6= 0 andhence W [x1, x2](t) 6= 0. Therefore, x1 and x2 form a fundamental set ofsolutions.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Example
Find the general solution to
x′ = Ax,
where A has eigenvalues λ = ±i and correspondingeigenvectors
v =
[2± i
1
].
Plot the phase portrait.
We can start by writing down one of our solutions in the formveλt .
xi =
[2 + i
1
]e it
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Example
Find the general solution to
x′ = Ax,
where A has eigenvalues λ = ±i and correspondingeigenvectors
v =
[2± i
1
].
Plot the phase portrait.
We can start by writing down one of our solutions in the formveλt .
xi =
[2 + i
1
]e it
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
xi =
[2 + i
1
]e it
We can use Euler’s formula to rewrite the e it term. That is,
xi =
[2 + i
1
](cos t + i sin t).
Separating real and imaginary terms gives us
xi =
[2 + i
1
](cos t + i sin t)
=
([21
]+ i
[10
])(cos t + i sin t)
=
([21
]cos t −
[10
]sin t
)+ i
([21
]sin t +
[10
]cos t
).
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
xi =
[2 + i
1
]e it
We can use Euler’s formula to rewrite the e it term. That is,
xi =
[2 + i
1
](cos t + i sin t).
Separating real and imaginary terms gives us
xi =
[2 + i
1
](cos t + i sin t)
=
([21
]+ i
[10
])(cos t + i sin t)
=
([21
]cos t −
[10
]sin t
)+ i
([21
]sin t +
[10
]cos t
).
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
xi =
[2 + i
1
]e it
We can use Euler’s formula to rewrite the e it term. That is,
xi =
[2 + i
1
](cos t + i sin t).
Separating real and imaginary terms gives us
xi =
[2 + i
1
](cos t + i sin t)
=
([21
]+ i
[10
])(cos t + i sin t)
=
([21
]cos t −
[10
]sin t
)+ i
([21
]sin t +
[10
]cos t
).
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
xi =
[2 + i
1
]e it
We can use Euler’s formula to rewrite the e it term. That is,
xi =
[2 + i
1
](cos t + i sin t).
Separating real and imaginary terms gives us
xi =
[2 + i
1
](cos t + i sin t)
=
([21
]+ i
[10
])(cos t + i sin t)
=
([21
]cos t −
[10
]sin t
)+ i
([21
]sin t +
[10
]cos t
).
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
xi =
[2 + i
1
]e it
We can use Euler’s formula to rewrite the e it term. That is,
xi =
[2 + i
1
](cos t + i sin t).
Separating real and imaginary terms gives us
xi =
[2 + i
1
](cos t + i sin t)
=
([21
]+ i
[10
])(cos t + i sin t)
=
([21
]cos t −
[10
]sin t
)+ i
([21
]sin t +
[10
]cos t
).
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
xi =
([21
]cos t −
[10
]sin t
)+i
([21
]sin t +
[10
]cos t
)It has been shown that Re(xi) and Im(xi) form a fundamentalset of solutions to x′ = Ax. Therefore, the general solution is
x = c1
([21
]cos t −
[10
]sin t
)+c2
([21
]sin t +
[10
]cos t
)or more compactly
x = c1
[2 cos t − sin t
cos t
]+ c2
[2 sin t + cos t
sin t
].
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
xi =
([21
]cos t −
[10
]sin t
)+i
([21
]sin t +
[10
]cos t
)It has been shown that Re(xi) and Im(xi) form a fundamentalset of solutions to x′ = Ax. Therefore, the general solution is
x = c1
([21
]cos t −
[10
]sin t
)+c2
([21
]sin t +
[10
]cos t
)or more compactly
x = c1
[2 cos t − sin t
cos t
]+ c2
[2 sin t + cos t
sin t
].
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
xi =
([21
]cos t −
[10
]sin t
)+i
([21
]sin t +
[10
]cos t
)It has been shown that Re(xi) and Im(xi) form a fundamentalset of solutions to x′ = Ax. Therefore, the general solution is
x = c1
([21
]cos t −
[10
]sin t
)+c2
([21
]sin t +
[10
]cos t
)or more compactly
x = c1
[2 cos t − sin t
cos t
]+ c2
[2 sin t + cos t
sin t
].
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
x = c1
[2 cos t − sin t
cos t
]+ c2
[2 sin t + cos t
sin t
]Recall that our eigenvalues were λ = ±i . The imaginary part of the eigenvalue leadsto rotations in our phase portrait due to the sines and cosines in our solution. Inaddition, since the real part of our eigenvalues is zero, the trajectories will not tend tothe origin or infinity as t →∞. Instead, they form closed orbits around the origin.This kind of critical point is known as a center. Since the trajectories don’t tend tothe origin as t →∞, we can’t call it asymptotically stable. Instead, we just call itstable since a particle that starts near the origin will remain close to the origin. Todetermine which direction the spiral is spinning, just evaluate x′ = Ax at a point.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
λ1 6= λ2 λ1 6= λ2 λ1 6= λ2
Both positive Both negative Opposite signs
Node Node SaddleUnstable Asymptotically stable Unstable
λ1 = 0 λ1 = 0λ2 > 0 λ2 < 0
Line of Eq. Pts. Line of Eq. Pts.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
λ = α± iβ λ = α± iβ λ = α± iβα < 0 α = 0 α > 0
Spiral sink Center Spiral sourceAsymptotically stable Stable Unstable
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations
Example
Consider the system of DE’s
x′ =
[2 −5α −2
]x.
(a) Determine the eigenvalues in terms of α.
(b) Find the critical value or values of α where the qualitativenature of the phase portrait for the system changes.
(c) Draw a phase portrait for a value of α slightly below, andanother value slightly above, each critical value.
(Gary Marple) October 2nd, 2017 Math 216: Introduction to Differential Equations