dt 1 2 f t f y dy b f b f y ̸0 - UHgorb/files/2011fall3321/_week03class.pdf3. ODEs. ODE: y ′ = f...

WEEK 3.SEPARABLE ODEs.

We write a first order ODE:

y′ = f(t, y) (1)

Def. The first order ODE (1) is SEPARABLE if and only if

f(t, y) = f1(t)f2(y).

Ex. f(t, y) =t

y⇒ f1(t) = t, f2(y) =

1

y

THM. If f1, f2 are continuous fns on some interval (a, b) and f2(y) ̸= 0on (a, b) then solutions of

dy

dt= f1(t)f2(y) (2)

are defined by ∫dy

f2(y)=

∫f1(t)dt+ C. (3)

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Ex.


EMCF03a.Q1–Q2

Q1. Is the equation y′ = tey−t separable?

(a) Yes

(b) No

Q2. Which of the following equations is separable?

(a) y′ =y + x

yx

(b) y′ = sin(yx)

(c) y′ = yex + y

(d) y′ = ey + ex


Ex.


EMCF03a.Q3

Q3. Which of the following equations is both linear and separable?

(a) y′ = y + 2x

(b) y′ =x

y + 2

(c) y′ =y2 + 1

y

(d) y′ = y + 2y


HOMOGENEOUS FUNCTIONS and ODEs

Def. A function f(t, y) is called homogeneous of degree n if it satisfies

f(κt, κy) = κnf(t, y)

for all every κ > 0, t ̸= 0, y ̸= 0.

Ex. f(t, y) =1

t2 + y2

f(t, y) = lnt

y

f(x, y) = 2x3 − 3x2y + 2xy2 − y3

f(x, y) =√x2 + y2

f(x, y) = sinx, f(t, y) = t2 + y are NOT HOMOGENEOUS


EMCF03a.Q4–Q5

Q4. Which of the following functions is homogeneous?

(a) f(t, y) =y + t

ty

(b) f(x, y) = arctan(yx)

(c) f(t, y) = yet + y

(d) f(x, y) =2xy

x2 + y

Q5. Determine the degree of the following homogeneous function:

f(x, y) =2y − x

x2.

(a) n = 2

(b) n = 1

(c) n = 0

(d) n = −1


Def. The first order equation

y′ = f(t, y)

is called homogeneous if the function f is a homogeneous function ofdegree 0.

EMCF03a.Q6

Q6. Which of the following equations is homogeneous?

(a) y′ =x2 − xy

xy − 1

(b) y′ =x− y

x+ y

(c) y′ = exy

(d) y′ =x2

3x+ y


Solving Homogeneous Equations

Given y′ = f(t, y) with f homogeneous of degree 0.By the change of dependent variable z =

y

twe convert it to a separable:

y = zt → y′ = z + z′t & f(t, y) = f(t, zt) = f(1, z) →

z + z′t = f(1, z) → z′ =f(1, z)− z

t separable

Ex.


.



Def. The first order equation y′ = f(t, y) is called a Bernoulli equationif it can be written in the form

y′ + p(t)y = q(t)yκ,

where p and q are continuous functions on some interval (a, b) andκ ̸= 0, 1 is a real number.

EMCF03a.Q7

Q7. Which of the following equations is Bernoulli’s?

(a) y′ + y2 = 0

(b) yy′ = exy

(c) y′ = xy

(d) xy′′ + y = exy2


Solving Bernoulli Equations

Multiplying the Eq. by y−κ and making change of variable z = y1−κ weobtain a linear Eq.:

z′ = (1− κ)y−κy′ & y−κy′ + p(t)y1−κ = q(t) →

1

1− κz′+p(t)z = q(t) or z′+(1−κ)p(t)z = (1−κ)q(t) linear

Ex.


.


APPLICATIONS

I. Constructing the ModelII. Analysis of the ModelIII. Comparison with Experiments or Observation

Major steps in constructing a model:

1. Identify independent and dependent var’s and assign letters torepresent them

2. Choose units of measurement

3. Articulate basic principle underlying or governing the investigatedproblem

4. Express this principle or law in terms of the chosen var’s

5. Make sure each term in the Eq. has the same units!

6. Construct initial condition

Ex. Suppose an object is falling in the atmosphere near sea level. As-sume only gravity and air resistance, which is proportional to the ob-ject’s velocity, act on the body.Formulate a DE that describes this motion


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