Math 6400: Homework Problems Gregor Kovaˇ ciˇ c 1. Consider the initial-value problem y 0 = f (x)g(y), y(x 0 )= y 0 , (1) where the functions f (x) and g(y) are continuous near x 0 and y 0 , respectively. Show that (i) If g(y 0 ) 6= 0, then there exists a unique solution of (1). (ii) If g(y 0 ) = 0, and g(y) 6= 0 for all y close enough to y 0 , then there exists a unique solution of (1) precisely when the integral Z y y 0 dη g(η) diverges. Find the solution in each case. 2. Consider the differential equation y 0 = p |y|. Find all of its possible solutions that pass through the point (x 0 , 0), and discuss your findings. 3. Consider Riccati’s equation y 0 + g(x)y + h(x)y 2 = k(x). (2) (i) If y = φ(x) is any solution of (2), then show that the substitution y = φ(x)+1/z will lead to its general solution. Write this solution down explicitly in terms of quadratures (i.e., integrals). 1
Transcript
Math 6400: Homework Problems
Gregor Kovacic
1. Consider the initial-value problem
y′ = f(x)g(y), y(x0) = y0, (1)
where the functions f(x) and g(y) are continuous near x0 and y0, respectively. Show that
(i) If g(y0) 6= 0, then there exists a unique solution of (1).
(ii) If g(y0) = 0, and g(y) 6= 0 for all y close enough to y0, then there exists a uniquesolution of (1) precisely when the integral∫ y
y0
dη
g(η)
diverges.
Find the solution in each case.
2. Consider the differential equation
y′ =√|y|.
Find all of its possible solutions that pass through the point (x0, 0), and discuss your findings.
3. Consider Riccati’s equation
y′ + g(x)y + h(x)y2 = k(x). (2)
(i) If y = φ(x) is any solution of (2), then show that the substitution y = φ(x) + 1/z willlead to its general solution. Write this solution down explicitly in terms of quadratures(i.e., integrals).
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(ii) Deduce from (i) that if φ1(x), φ2(x), and φ3(x) are three particular solutions of (2),then its general solution is given by
y − φ2(x)
y − φ1(x)= C
φ3(x)− φ2(x)
φ3(x)− φ1(x),
where C is an arbitary constant.
(iii) Show that the substitution
y =u′
h(x)u
will transform (2) into a linear second order equation.
(i) Find a parametric solution (x(p), y(p)) of (3) in terms of the parameter p = y′. Differ-entiating (3) once will help.
(ii) Find a family of straight-line solutions of (3).
(iii) Show that the solution found in (i) is the envelope of the family of straight linesfound in (ii). (First show that the envelope of the family of curves given by an equa-tion F (x, y, c) = 0 is obtained by eliminating c from this equation and the equation∂cF (x, y, c) = 0.)
5. Consider the system
E = P − αE, P = E − βP, D = −γD, (4)
where the overdot denotes differentiation with respect to the time-variable t, and α, β, andγ are non-negative parameters.
(i) Find the general solution of the system (4).
(ii) Find the stable and unstable subspaces of the origin, that is, the sets of all solutions ofsystem (4) that approach the origin for t→∞ and t→ −∞, respectively. What kindof geometric objects are these subspaces? How do they change with the parameters α,β, and γ?
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6. Let the function k(x, t, z) be continuous, and let its partial derivative ∂zk(x, t, z) becontinuous and uniformly bounded for 0 ≤ t ≤ x ≤ a and all real z. Also, let the functiong(x) be continuous for 0 ≤ x ≤ a. Show that the Volterra integral equation
u(x) = g(x) +
∫ x
0
k(x, t, u(t)) dt
possesses exactly one continuous solution in the interval 0 ≤ x ≤ a.
7. Consider the equation x = Ax, where x is an n-dimensional real vector, and A is a givenn × n real matrix. Use Picard iterations to show that the general solution of this equationis x = eAtx0, where x0 is any constant vector, and
eAt =∞∑n=0
tn
n!An.
8. Investigate how far solutions of the initial-value problem
x = x2, x(t0) = x0
can be extended.
9. In this exercise, you will show a theorem on extension of solutions by filling in the stepsof the following outline:
(i) Define a function f(x, y) to satisfy a local Lipschitz condition in the set D ⊂ R2 if for everypoint (x0, y0) ∈ D, there exists a neighborhood U = U(x0, y0) and a number L = L(x0, y0),such that the function f satisfies the Lipschitz condition
|f(x, y)− f(x, y)| ≤ L |y − y| (5)
on the intersection D ∩ U .
Show that if the set D is open and if f ∈ C(D) has a continuous derivative fy in D, then fsatisfies a local Lipschitz condition in D.
(ii) Show that if D is open and f ∈ C(D) satisfies a local Lipschitz condition in D, then theinitial-value problem
y′ = f(x, y), y(ξ) = η (6)
locally has a unique solution for every (ξ, η) ∈ D. In other words, in some small enoughneighborhood of every ξ, there exists a unique solution.
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(iii) Let f be defined in D and let Φ = φαα∈A be a family of solutions for the initial-valueproblem (6) in the respective intervals Jα, such that
φα(x) = φβ(x) for x ∈ Jα ∩ Jβ (α, β ∈ A). (7)
Show that there exists a unique solution φ of (6) in the interval J = ∪α∈AJα with φ|Jα = φαfor all α ∈ A.
An immediate consequence is that if the initial-value problem (6) has at least one solution,and the uniqueness statement (7) holds for any two solutions, then there exists a solution of(6) that cannot be extended. All other solutions are restrictions of this solution.
REMARK: Note that ξ ∈ Jα for all α.
HINT: First show φ is uniquely defined. Then show that φ is indeed a solution in J .
(iv) Let D ⊂ R2 and f ∈ C(D). If φ is a solution of the differential equation y′ = f(x, y) onthe interval ξ ≤ x < b, which is entirely contained in a compact set A ⊂ D, then show thatφ can be extended as a solution to the closed interval [ξ, b].
HINT: Show that boundedness of f on A implies the existence of the limit limx→b− φ(x).Define φ(b) to be this limit, and show that the thus-obtained extended function φ is left-differentiable at b and satisfies the differential equation.
(v) Let D ⊂ R2 and f ∈ C(D). Let φ be a solution of the differential equation y′ = f(x, y)on the interval [ξ, b], let ψ be a solution on the interval [b, c], and let φ(b) = ψ(b). Show that
u(x) =
φ(x) for ξ ≤ x ≤ b
ψ(x) for b < x ≤ c
is a solution in the interval [ξ, c].
HINT: Compare left-hand and right-hand derivatives.
(vi) Let D ⊂ R2 be open, and let f ∈ C(D) satisfy a local Lipschitz condition in D. Showthat for every (ξ, η) ∈ D, the initial-value problem (6) has a solution φ which cannot beextended, and which both on the right and on the left of ξ approaches the boundary of Darbitrarily closely. Furthermore, show that the solution φ is uniquely determined, that is,all other solutions of (6) are restrictions of φ.
REMARK: The statement that φ on the right of ξ approaches the boundary of D arbitrarilyclosely is defined as follows: If G is the closure of the graph of φ, and G+ is the subset ofthe points (x, y) ∈ G with x ≥ ξ, then G+ is not a compact subset of D.
An equivalent, but more intuitive, formulation is the following:
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The solution φ exists on the right of ξ in some interval ξ ≤ x < b (with b = ∞ permitted),and one of the following statements holds:
(a) b =∞; the solution exists for all x ≥ ξ.
(b) b <∞, lim supx→b− |φ(x)| =∞; the solution “blows up.”
(c) b < ∞, lim supx→b− ρ(x, φ(x)) = 0, where ρ(x0, y0) is the distance of the point (x0, y0)from the boundary of D; the solution “approaches the boundary arbitrarily closely.”
In fact, the above definition states that G+ is either unbounded (case (a) or (b)), or elsebounded and contains boundary points of D.
HINT: For uniqueness, assume that the solution “splits” at some point x0, say x0 > ξ. Derivea contradiction with (ii).
For existence use (ii) and the fact that by the uniqueness that you just proved, (7) holds.Use the consequence of (iii) to show the existence.
For the approach to the boundary of D assume that G+ is a compact subset of D, and thatφ exists on a finite interval, either ξ ≤ x < b or ξ ≤ x ≤ b. Use either (iv) or (ii) and (v) toextend φ.
10. Let the functions k(x, t, y;λ), g(x;λ), and α(λ) be twice continuously differentiable fora ≤ x ≤ b, a ≤ t ≤ b, all y ∈ Rn, and λ ∈ K0, where K is a compact subset of Rm and K0
is its interior. Moreover, let a ≤ α(λ) ≤ b on K. If all the first and second derivatives of thefunctions k(x, t, y;λ), g(x;λ), and α(λ) are uniformly bounded in their respective domains,show that the solution of the integral equation
y(x;λ) = g(x;λ) +
∫ x
α(λ)
k(x, t, y(t;λ);λ) dt
is continuously differentiable.
HINT: Take the formal derivative of this integral equation with respect to a component,say λ′, of λ, and consider the resulting pair of equations for y and v = ∂λ′y on the producty − v-space. Show that this pair satisfies a Lipschitz condition in (y, v).
11. Let x = φ(t, x0) be a solution of the initial-value problem
x = f(x, t), x(t0) = x0.
Show that, under the appropriate smoothness hypotheses, its derivative v = ∂x0φ(t, x0)satsifies the first variation equation and the initial condition
v = ∂xf(x, t)v, v(t0) = I,
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respectively, where I is the identity matrix.
12. If the right-hand side of the differential equation x = f(x, t) is r-times continuouslydifferentiable, show that its solution x = φ(t, x0) with φ(t0, x0) = x0 is r-times continuouslydifferentiable with respect to t, t0, and x0.
HINT: For x0, use induction on r. For t and t0 also use the integral equation.
13. A family of functions F = f is equicontinuous on an interval I, if given any ε > 0there exists a δ > 0 such that |f(t1) − f(t2)| < ε whenever |t1 − t2| < δ and f ∈ F . ProveArzela-Ascoli’s lemma
Theorem 1 Let I be a compact interval, and let F = f be an infinite, uniformly bounded,and equicontinuous family of functions. Then F contains a sequence fn | n = 1, 2, . . .,which is uniformly convergent on I.
HINT: Let rk | k = 1, 2, . . . be all the rationals in I enumerated in some order. Show thatthere exist functions fnk ∈ F such that the sequence fnk converges at r1, . . . , rk, and thatyou can take fn = fnn as the desired unformly convergent subsequence.
14. Prove Peano’s existence theorem
Theorem 2 Let the function f(t, x) be continuous and bounded on the strip defined by 0 ≤t ≤ 1, −∞ < x < ∞. Then there exists at least one continuously differentiable solution ofthe initial-value problem
x = f(t, x), x(0) = x0 (8)
on the interval 0 ≤ t ≤ 1.
HINT: Fix n. For i = 0, . . . , n put ti = i/n. Let φn be a continuous function on 0 ≤ t ≤ 1such that φn(0) = x0,
φn(t) = f(ti, φn(ti)) if ti < t < ti+1,
and put∆n(t) = φn(t)− f(t, φn(t)),
except at the points ti, where ∆n(t) = 0. Then
φn(t) = x0 +
∫ t
0
[f(τ, φn(τ)) + ∆n(τ)] dτ.
Choose M so that f < M . Verify the following assertions:
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(a) |φn| ≤ M , |∆n| ≤ 2M , ∆n Riemann integrable, and |φn| ≤ |x0| + M = M1, say, on0 ≤ t ≤ 1, for all n.
(b) φn is equicontinuous on 0 ≤ t ≤ 1, since |φn| ≤M .
(c) Some φnk converges to some φ, uniformly on 0 ≤ t ≤ 1.
(d) Since f is uniformly continuous on the rectangle 0 ≤ t ≤ 1, |x| ≤M1,
f(t, φnk(t))→ f(t, φ(t))
uniformly on 0 ≤ t ≤ 1.
(e) ∆n(t)→ 0 uniformly on 0 ≤ t ≤ 1 since
∆n(t) = f(ti, φn(ti))− f(t, φn(t))
for ti < t < ti+1.
(f) Hence
φ(t) = x0 +
∫ t
0
f(τ, φ(τ)) dτ.
This φ is the solution of the given problem.
15. Let q = q1 + iq2 and p = p1 + ip2, with qj, pj ∈ R for j = 1, 2, and i2 = −1. Considerthe complex Duffing equation
q = p, p = q
(K − 1
2|q|2), (9)
where K is an arbitary constant.
(i) Show that system (9) is Hamiltonian with the Hamiltonian function
H =1
2|p|2 +
1
2
(K − 1
2|q|2)2
,
and that equations (9) are of the form
q = 2∂H
∂p∗, p = −2
∂H
∂q∗,
where ∗ denotes complex conjugation.
(ii) Show that the function
J =1
2i(q∗p− qp∗) = q1p2 − q2p1
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is a first integral of system (9). Show that the Hamiltonian H is invariant under the groupof transformations gθ given by the solutions of the system
dq
dθ= 2
∂J
∂p∗= iq,
dp
dθ= −2
∂J
∂q∗= ip.
Show that if q(θ = 0) = Q ∈ R is the initial condition of the first equation in this system,then the initial condition of the second equation must be p(θ = 0) = P + iJ/Q for someP ∈ R.
(iii) Show that the coordinate change
q = Qeiθ, p =
(P + i
J
Q
)eiθ,
with Q, P and θ ∈ R, brings the Hamiltonian H into
H =1
2
(P 2 +
J2
Q2
)+
1
2
(K − 1
2Q2
)2
,
and equations (9) into
Q = P, P = Q(K − 1
2Q2)− J2
Q3, θ =
J
Q2, J = 0, (10)
and that this system is Hamiltonian with the Hamiltonian function H. Notice that system(10) is decoupled in the following sense: Since J is constant, the first two equations are aplanar system for the variables Q and P . Once this system is solved, the angle θ can becalculated by quadrature (i.e., integration).
(iv) Sketch the phase portraits of the Q − P system for positive and negative K, and zeroand nonzero J . What geometric objects in the full q − p space do the periodic orbits in theQ− P plane correspond to?
(v) There are two separatrix loops in the Q − P phase plane for K > 0 and J = 0. Findthe solutions on these two loops. Show that, in the full q − p phase space, these solutionscorrespond to the solutions
q = 2√K sech(
√Kt)eiθ0 , p = −2K sech(
√Kt) tanh (
√Kt)eiθ0 ,
with constant 0 ≤ θ0 ≤ 2π. What geometric object do these solutions trace out?
Remark: The remarkable connections between equations (9), the integral J , the groupgθ, and the transformation that leads to system (10) are not accidental. In fact, almostevery multi-dimensional conservative mechanical system solved to date is solvable because
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of additional symmetries, which, in turn, generate conserved quantities and make it possibleto reduce the system to quadratures.
16. Reduction of order: Let y1(x) be a nonzero solution of the second order linear equation
y′′ + p(x)y′ + q(x)y = 0. (11)
Show that the ansatz y = y1(x)u will lead you to an equation for u which can be solved byquadrature, and hence to the general solution of (11).
17. Euler’s Equation readsx2y′′ + a1xy
′ + a0y = 0. (12)
(i) Show that if y(x) is a solution of (12), then so is y(−x). Deduce that it is enough toconsider solutions of (12) for x > 0.
(ii) Show that assuming y = xr in (12) leads to a quadratic equation for r. Find the formof two linearly independent solutions of (12) if the roots of this quadratic equation are realr1 6= r2.
(iii) Find the form of two linearly independent solutions of (12) if the roots of the quadraticequation for the exponent r are complex conjugate r1 = λ+ iµ, r2 = λ− iµ.
(iv) Use reduction of order or the Wronskian to find the solution of (12) if the quadraticequation for the exponent r has two equal real roots.
(v) Euler’s equation (12) is a perfect counter-example used to show what all can go wrongwith the solutions of a differential equations at points where the existence theorem does nothold. Where does Euler’s equation have such points?
(vi) How far can the solution of Euler’s equation with the initial condition y(x0) = y0, withx0 > 0, be extended.
(vii) Consider the situations in which the exponents r1 and r2 corresponding to a particularEuler’s equation are: (a) r1 = 1, r2 = 2; (b) r1 = 1, r2 = −1; (c) r1 = r2 = 1; (d) r1 = i,r2 = −i; (e) r1 = 1 + i, r2 = 1− i; (f) r1 = −1 + i, r2 = −1− i. Write down the solutions ineach case, draw their graphs, and discuss their behavior near the origin. Also, explain whatgoes wrong with the initial-value problem y(0) = y0 in each case.
18. Show that if A(t) is a continuous n×n matrix, Φ is a fundamental matrix of the system
x = A(t)x, (13)
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and C is a constant nonsingular matrix, then ΦC is another fundamental matrix of system(13). Show also that every fundamental matrix of (13) is of this form.
19. Let A† be the adjoint (complex conjugate and transposed) matrix of A. The system
x = −A†(t)x (14)
is the adjoint system to system (13). Show that if Φ is a fundamental matrix of (13), thenΨ is a fundamental matrix of (14) if and only if Ψ†Φ = C where C is a nonsingular constantmatrix.
20. Consider the equationx+ q(t)x = 0, (15)
where q(t) is a continuous function, in the phase plane (x, y = x). By following the outlinebelow, you will prove the Sturm oscillation and comparison theorems, which are useful foranalyzing the Sturm-Liouville eigenvalue problem.
(i) Show that trajectories of (15) intersect the ray x = 0, y > 0 at points where x isincreasing and the ray x = 0, y < 0 at points where x is decreasing along the trajectory.
(ii) Deduce from part (i) that for any two successive intersections of a solution with they-axis one occurs with y > 0 and the other with y < 0.
Let φ be the polar angle measured clockwise from the positive y-axis.
(iii) Use part (ii) to show that between any two successive intersections of a trajectory withthe y-axis, the angle φ increases by π along this trajectory.
(iv) Show the Sturm Oscillation Theorem: On the interval between two successive zerosof any solution of equation (15) there is a zero of any other solution.
HINT: Consider the polar angles φ = α(t) and φ = β(t). For any two linearly independentsolutions, α(t) 6= β(t) for all t. (Why?)
(v) Show
φ =q(t)x2 + y2
x2 + y2
(vi) Show the Sturm Comparison Theorem: Consider two equations of the form (15)
x+ q(t)x = 0, (16a)
x+Q(t)x = 0, (16b)
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and assume that Q(t) ≥ q(t). Then on the interval between any two successive zeros of anysolution of (16a) there is a zero of any solution of (16b).
HINT: First assume that Q(t) > q(t). Then use the result of part (v) to show that thepolar angle φ = A(t) corresponding to a solution of (16b) always grows faster than the angleφ = α(t) corresponding to a solution of (16a).
(vii) Show that the distance between any two sucessive zeros of (15) is
• not larger than π/ω if q(t) ≥ ω2 for all t,
• not smaller than π/Ω if q(t) ≤ Ω2 for all t.
In particular, if q(t) ≤ 0 for all t, then no solution of (15) except the identically zero solutioncan have more than one zero.
21. Consider now the Sturm-Liouville eigenvalue problem:
where λ is a constant parameter. The values of λ for which solutions of (17) that are notidentically zero exist are called the eigenvalues. The corresponding solutions are called theeigenfunctions.
(i) Show that for any function q(t) that is smooth on the interval [0, l], (17) has an infiniteset of eigenvalues. The corresponding eigenfunctions may have an arbitrarily large numberof zeros on this interval.
HINT: Consider the solution of (17) with initial condition x(0) = 0, x(0) = 1, and let φ =α(t, λ) be its polar angle. Use the Sturm Comparison Theorem to show that α(l, λ)→∞ asλ→∞. Conclude that there exists an infinite set of eigenvalues λn for which α(l, λn) = nπ.
(ii) Show that
limn→∞
λnn2
=(πl
)2
.
22. Let x = φ(t) be a periodic solution of the system
x = f(x), x ∈ Rn, f ∈ Cr(Rn) for some r ≥ 1. (18)
Let Σ be any local, (n− 1)-dimensional surface, called the Poincare section, transverse (i.e.,non-tangent) to the orbit O of the solution φ(t). Let P : Σ→ Σ be the Poincare map, that
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is, for any appropriate point p ∈ Σ, P (p) is the first point in which the orbit through pintersects Σ again for some t > 0.
(a) Draw a sketch of the orbit O, the Poincare section Σ, and the Poincare map P . Showthat P is well defined and continuous on some small enough subset U ⊂ Σ.
(b) Show that any two Poincare maps P and P ′ are differentiably conjugate, that is, thereexist a diffeomeorphism σ(P, P ′) (i.e., both σ and its inverse σ−1 are continuously differen-tiable) such that P = σ(P, P ′) P ′ σ−1(P, P ′). Just what is this σ(P, P ′)?
HINT: You can assume that the solution of a differential equation is as smooth as its right-hand side in time, initial conditions, and parameters.
(c) Consider the linearizationu = Df(φ(t))u
of equation (18) about the solution x = φ(t). Show that (n− 1) Floquet multipliers of thislinear system are equal to the eigenvalues of any Poincare map P : Σ→ Σ, linearized aboutthe fixed point pO on Σ that is the intersection of Σ with the periodic solution x = φ(t).Show that the remaining multiplier equals 1. What is the eigenvector that corresponds tothis last multiplier?
23. Consider the system
A1 = −iβ1A1 + iµ
4|A1|2A1 − iA∗1A2, (19a)
A2 = −iβ2A2 + iµ2
2|A2|2A2 −
i
2A2
1, (19b)
where A1 and A2 are complex, i2 = −1, ∗ denotes the complex conjugate and | · | denotesthe complex modulus.
(i) Show that the system (19) possesses an invariant plane with A1 = 0. Find the generalsolution A2 = a(t) in this plane.
(ii) Linearize the system (19) about the solution a(t), that is, assume the ansatz
(A1(t), A2(t)) = (u1(t), a(t) + u2(t)),
substitute this ansatz into (19), and neglect higher powers of u1(t) and u2(t).
(iii) Show that a time-dependent coordinate change transforms the resulting linear systeminto a linear system with constant coefficients, and thus solve it.
(iv) Compute explicitly the Floquet multipliers corresponding to the periodic solutions ofsystem (19) in the plane A1 = 0.
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(v) Exhibit explicitly the solutions of the linearization of system (19) around the periodicsolutions in the plane A1 = 0 in the form implied by the Floquet theorem, that is, ψ(t)eRt,where ψ(t) is periodic and R is a constant matrix.
24. Consider the system
a =[λ+ Aa(a, b, µ)a+ Ab(a, b, µ)b
]a, (20a)
b =[−λ+Ba(a, b, µ)a+Bb(a, b, µ)b
]b, (20b)
defined for |a|, |b| < δ for some positive δ. Here, λ > 0 and µ are constant parameters.Assume that all the functions in (20) are uniformly bounded in all their arguments.
(a) Show that the lines a = 0 and b = 0 are both invariant under the flow of system (20),and that the origin a = b = 0 is an equilibrium point.
(b) Let κ be any number satisfying 0 < κ < λ, and let and δ be sufficiently small. Considera solution starting at t = 0 with the initial conditions a = a0, b = δ, and let T be the timethat it takes this trajectory to reach one of the lines a = δ or a = −δ. Show that there existtwo constants Ca and Cb such that for any 0 < t < τ < T , we have
|a(τ)| ≥ Caeκ(τ−t)|a(t)|, |b(τ)| ≤ Cbe
−κ(τ−t)|b(t)|.
Just how small must δ be?
HINT: Show the second inequality; the first is shown in the same way in backward time. Toshow the second inequality, first use the variation of constants formula to eliminate the linearpart, then use the uniform boundedness of the functions that are left on the right-hand side.
25. (i) Show that two linearly independent power series solutions of Airy’s equation
w′′ − zw = 0
about the point z = 0 are given by
w1(z) =∞∑n=0
z3n
32nn! Γ(n+ 2
3
) , w1(z) =∞∑n=0
z3n+1
32nn! Γ(n+ 4
3
) .Recall that
Γ(α) =
∫ ∞0
xα−1e−x dx
for α > 0, and is defined by analytic continuation for all other values of α, except negativeintegers, where it has simple poles.
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(ii) Show that the Airy function
Ai(z) =1
2πi
∫Cezζ−ζ
3/3dζ,
where C is the path starting at ∞ with the argument −2π/3 and ending at ∞ with theargument 2π/3, satisfies Airy’s equation and express it in terms of these solutions. Whatyou should get is
Ai(z) =∞∑n=0
z3n
32n+ 23n! Γ
(n+ 2
3
) − ∞∑n=0
z3n+1
32n+ 43n! Γ
(n+ 4
3
) .HINT: Express Ai(0) and Ai′(0) in terms of the two power-series solutions at z = 0. Tocompute these two values, first deform the integration path to run along the imaginary axis(why can you do that?), then change the variable so it runs along the real axis. In computingAi(0), split the resulting integral in two and change one of the two integrals to get the sumof the two integrals
∫∞0e±it
3/3dt. Show that you can rotate the integration paths to becomethe half-rays emerging from the origin at the angle ±π/6, which will give you real integrals.Do something similar for Ai′(0). Express all these integrals in terms of the Gamma function,and, just before the end, use the well-known formula
Γ(α)Γ(1− α) =π
sin πα.
26. Show that for non-integer values of ν, two linearly independent solutions of Bessel’sequation
x2y′′ + xy′ + (x2 − ν2)y = 0
are given by Jν(x) and J−ν(x), where
Jν(x) =∞∑n=0
(−1)n
n! Γ(n+ ν + 1)
(x2
)2n+ν
.
Use the property that Γ(α + 1) = αΓ(α) in the series for Jν(x).
If ν is an integer, use the fact that 1/Γ(−k) = 0 for non-negative integer k to show thatJν(x) = (−1)νJ−ν(x), so that these two functions give only one linearly independent solution.Show that when ν is a non-negative integer, another independent solution is given by
Jν(x) =2Jν(x) logx
2−
ν−1∑k=0
(ν − k − 1)!
k!
(x2
)2k−ν−(x
2
)ν 1
ν!
ν∑k=1
1
k−
∞∑k=1
[k+ν∑m=1
1
m+
k∑m=1
1
m
](−1)k
k!(ν + k)!
(x2
)2k+ν
.
14
Remark: The standard second solution in this case is the limit of the the Neumann function,
Yν(x) =Jν(x) cos(νπ)− J−ν(x)
sin(νπ),
as ν tends to the appropriate non-negative integer. It is given by the formula
Yν(x) =1
π[2γJν(x) + Jν(x)] ,
where
γ = limn→∞
(n∑k=1
1
k− log n
)is the well-known Euler-Mascheroni constant.
27. Show that the equation
w′′ −(
1 +1
2z
)w′ = 0
has formal solutions of the form
zα∞∑
n=−∞
anzn
for any α, but that only special α give a convergent series and hence genuine solutions.
28. Use the transformation method on the hypergeometric equation to find two independentsolutions which are functions of 1/z valid in |z| > 1.
29. (i) Find the restrictions on p(z) and q(z) at a regular singular point of the equationw′′+ p(z)w′+ q(z)w = 0 if the exponents (i.e., roots of the indicial equation) are to be 0 and1/2.
(ii) Find the most general form of the second-order equation with regular singular points at0, 1, k, ∞ and exponents 0, 1/2 at 0, 1, k (but not ∞).
30. (a) Set ζ = kz, b = k in the series solutions z−aF (1/z) in Problem 28, and show howthe convergence is lost as k →∞.
(b) Using one of the identities, the second solution for |z| > 1 can be written
z−b(
1− 1
z
)c−a−bF
(c− a, 1− a, b− 1− a;
1
z
).
15
Carry out the same limiting process on this solution. The resulting series is an asymptoticexpansion.
31. Lame’s equation, which has for regular singular points at 0, 1, k, and ∞, can be takenin the form
d2w
dζ2+
1
2
(1
ζ+
1
ζ − 1+
1
ζ − k
)dw
dζ+
A+Bζ
ζ(ζ − 1)(ζ − k)w = 0.
Find the confluent form as k → ∞, and show that the confluent form includes a transfor-mation of Mathieu’s equation
d2w
dz2+ 4
(α + β cos2 z
)w = 0.
32. Classify singular points of the equation
d2w
dz2+ z
dw
dz+ µw = 0, µ = constant,
and find the first term of the asymptotic expansion for each of the two solutions as z →∞.
33. Find an approximate solution of the equation
d2y
dx2+ 256e4xy = 0 in x > 0,
with initial conditions y(0) = 0, y′(0) = 1. Find the approximate position and magnitude ofthe first maximum of y(x).
34. For water waves propagating in two space dimensions (x, z) on water of depth h(x), anapproximate equation for the height of the waves η(x, z, t) is
∂2η
∂t2= gh(x)
(∂2η
∂x2+∂2η
∂z2
)+ gh′(x)
∂η
∂x.
This applies very approximately to long waves on a beach; the shoreline is x = 0 and thedepth h(x) increases out to sea. Consider waves propagating mainly along the shoreline (i.e.,in the z direction) of the form
η(x, z, t) = f(x)eik(z−ct),
where k and c are constant. (a) Show via the WKB method that the waves can be trappedby a transition region at a value x offshore. (b) In the case h(x) = αx, find approximatesolutions for f(x) on the two sides of the transition region and connect them.
16
35. Derive a transformation of variables to put the equation
d2y
dx2+[λ2q(x) + r(x)
]y = 0
into the formd2η
dξ2+[λ2ξ +R(ξ)
]η = 0.
(This transformation is motivated by the idea of getting the solution close to the Airyfunction everywhere, not just at transition points.)
36. Consider a vector field f(x) = (f1(x1, x2), f2(x1, x2)) on R2. Its divergence ∇ · f(x) isdefined as
∇ · f(x) =∂f1(x1, x2)
∂x1
+∂f2(x1, x2)
∂x2
.
Prove the following
Theorem (Bendixson’s criterion) If on a simply connected region D ⊂ R2, the divergence∇ · f(x) of the vector field f(x) does not vanish identically and does not change sign, thenthe differential equation x = f(x) has no closed orbits in D.
Recall that a simply connected region is one without holes.
HINT: Use Green’s theorem.
37. The Gronwall inequality: Let φ, ψ, and χ be real-valued continuous (or piecewisecontinuous) functions on a real t-interval I: a ≤ t ≤ b. Let χ(t) > 0 on I, and suppose fort ∈ I that
φ(t) ≤ ψ(t) +
∫ t
a
χ(s)φ(s) ds.
Prove that on I
φ(t) ≤ ψ(t) +
∫ t
a
χ(s)ψ(s) exp
(∫ t
s
χ(u) du
)ds.
HINT: Let R(t) =
∫ t
a
χ(s)φ(s) ds and show that R− χR ≤ χψ.
38. Show the following
Theorem: Letx = Ax+ f(x), (21)
17
where A is a real constant matrix with the eigenvalues all having negative real parts. Let fbe real, continuous for small x and
lim‖x‖→0
‖f(x)‖‖x‖
= 0.
Then the identically zero solution is asymptotically stable.
HINT: Use variation of constants to derive an integral equation for the solution φ of (21)with given φ(0). The estimate
‖eAt‖ ≤ Ke−σt
which holds for some K, σ > 0, and all t ≥ 0 (why?), implies an integral inequality for‖φ(t)‖. Now, given any ε > 0, there exists a δ > 0 such that ‖f(x)‖ ≤ ε‖x‖/K for ‖x‖ < δ.Conclude that so long as ‖φ(t)‖ < δ, one must have
eσt‖φ(t)‖ ≤ K‖φ(0)‖+ ε
∫ t
0
eσs‖φ(s)‖ ds.
Use the Gronwall inequality to conclude that
‖φ(t)‖ ≤ K‖φ(0)‖e−(σ−ε)t.
Choosing appropriately small (how small?) ε and ‖φ(0)‖ leads to the conclusion of the proof.
39. Consider the differential equation
x = f(x), x ∈ Rn, f(0) = 0.
Let V (x) be a real, continuously differentiable function defined in a small neighborhood Uof 0 ∈ Rn.
Prove the following
Theorem If
(i) V (0) = 0 and V (x) > 0 in U − 0, and
(ii) V (x) ≤ 0 in U − 0,
then 0 is a stable equilibrium. Moreover, if
(iii) V (x) < 0 in U − 0,
18
then 0 is an asymptotically stable equilibrium.
Draw an appropriate figure for each of the two situations. The function V (x) is known asthe Lyapunov function.
HINT: Given ε > 0, let k(ε) = min‖x‖=ε V (x). Show that δ(ε) required in the definitionof stability is given by δ(ε) = minx|V (x)=k(ε) ‖x‖. For the asymptotic stability, assume thecontary, namely, that for every ε > 0 there exist δ(ε), λ(ε) > 0 and a solution xε(t) suchthat xε(0) < δ(ε) and that xε(t) ≥ λ(ε) for all t > 0. Use the fact that V (xε(t)) ≤ −d(ε) < 0for some d(ε) to derive a contradiction with the fact that V (xε(t)) > 0 for all t > 0.
40. The following is an outline of the original stable-manifold theorem by Hadamard. Thistheorem appeared under the title “Sur l’iteration et les solutions asymptotiques des equationsdifferentielles” in Bull. Soc. Math. France, 29 (1901), pp. 224–228.
Consider the map (x, y) 7→ (x1, y1) of R2, given by
x1 = sx+ F (x, y), y1 = s′y + Φ(x, y), (22)
where s > 1 > s′, and the functions F (x, y) and Φ(x, y) are smooth at the origin and beginwith quadratic terms there.
(i) Consider a curve segment C that passes through the origin and is determined by a functiony(x), whose derivative is bounded from above by α and from below by −α for some α > 0for all small enough x. Show that, restricted to small enough x, the image C1 of C is of thesame type.
(ii) Let C ′ be a curve segment of the same type as C. Verify that y(x)− y′(x)→ 0 as x→ 0and |y(x)− y′(x)|/x < µ for some µ > 0.
Let then C1 and C ′1 be the images of C and C ′, respectively, and let µ1 be the analog for C1
and C ′1 of the number µ introduced in part (ii). In the next several steps, you will show thatµ1/µ can be taken as close to s′/s as we please if we restrict the domain to small enough x.To this end, let Y and Y ′ be the ordinates on C and C ′ corresponding to the same abscissa X,which we can assume positive with no loss of generality. Let y1 and y′1 be the correspondingordinates on C1 and C ′1, and let the preimages of the points (X, y1) and (X, y′1) be the points(x, y) and (x′, y′) on the curves C and C ′, respectively.
(iii) Choose any η > 0, and let x be small enough. Show that the assumptions about C andthe functions F and Φ imply the inequalities
|y| < αx (23)
and|X − sx| < η(x+ |y|), (24)
19
and that therefore x and x′ are both smaller than
X
s− η(1 + α).
(iv) If y′0 is the ordinate of the point on C ′1 with abscissa x, show that
|y′0 − y| < µx <µX
s− η(1 + α), (25)
|y′ − y′0| < α|x′ − x|. (26)
(v) Show that since the magnitudes of the derivatives of F and Φ are smaller than η,
You will now deduce the existence of the unstable manifold of the origin.
(vii) Let C ′ = C1. Then C ′1 = C2, the image of C under the second iterate of the map (22). Ifwe denote the image of C under the n-th iterate of the map (22) by Cn, then show that thedifference between the ordinates of the points on the curves Cn and Cn+1 with the abscissaX are smaller than µσnX, where σ = µ1/µ.
(viii) Deduce that the result of the previous paragraph implies the existence of a uniquelimiting curve K = limn→∞ Cn, which is independent of the choice of C, invariant under themap (22), and which passes through the origin, and is not tangent to the y axis there. Thex coordinates of points on K grow in absolute value under interations of the map (22).
(ix) Show that K is tangent to the x axis at the origin.
20
41. In class, we saw that the transcritical and pitchfork bifurcations only occur when specialconditions are imposed on the system under consideration. To find out what happens whenthese conditions are relaxed, consider the vector fields
x = ε+ µx± x2 and x = ε+ µx± x3,
where µ and ε are parameters. For each vector field, plot three bifurcation diagrams in theµ − x plane: one for ε > 0, one for ε = 0, and one for ε < 0. To sketch the equilibria,compute µ in terms of x rather than the other way around.
42. Consider local bifurcations of the families of Hamiltonian systems with one degree offreedom:
x = p, p = f(x, µ). (31)
(a) Show that each such system can be derived from the Hamiltonian function
H(x, p, µ) =1
2p2 + V (x, µ)
via the formulas
x =∂H(x, p, µ)
∂p, p = −∂H(x, p, µ)
∂x,
where V (x, µ) = −∫f(x, µ) dx.
(b) Show that all equilibria of system (31) lie on the x-axis, and that the eigenvalues ateach equilibrium occur in pairs ±λ.
(c) Consider the following six functions for f(x, µ):
µ± x2, µx± x2, µx± x3.
Describe how the equilibria of the correspoding systems (31) undergo the Hamiltonianversions of the saddle-note, transcritical, and pitchfork bifurcations, respectively. Inparticular, what is the linearization matrix at the bifurcation point? Show that thesebifurcations do not only create new equilibria or exchange their stability, but alsocreate new families of periodic orbits and new separatrix curves, or make the equilibriaexchange the separatrix loops that they possess and families of periodic orbits thatencircle them. Present three phase portraits for each function f(x, µ): one for µ > 0,one for µ = 0, and one for µ < 0.
43. Consider the system
E = P − αE, P = ED − βP, D = −EP − γ(D − µ).
21
Determine all the equilibria of this system and all their bifurcation points. Classify thebifurcations according to their type: saddle-node, transcritical, pitchfork, or Hopf.
44. Consider the Lorenz system
x = σ(y − x), y = ρx− y − xz, z = −βz + xy, (32)
where σ and β are fixed positive constants and ρ is a parameter. The point x = y = z = 0is always an equilibrium of this system. Show that it undergoes a bifurcation at ρ = 1. Letµ = ρ−1, add the equation µ = 0 to the system (32), and use the method for approximatingcenter manifolds with parameters to analyze this bifurcation.
45. For iterated maps, formulate the center manifold theorem and related results that weregiven in class for vector fields. You may assume that the map has the form
xn+1 = Axn + f(xn, yn), yn+1 = Byn + g(xn, yn),
where the matrices A and B have eigenvalues with moduli equal to 1 and different from1, respectively, and the functions f and g vanish at the origin together with their firstderivatives. Show that if the center manifold is given by the equation y = h(x), then thefunction h satisfies the equation
N (h(x)) = h(Ax+ f(x, h(x)))−Bh(x)− g(x, h(x)) = 0.
Use this equation to approximate the center manifold and the dynamics on it for the mapping
xn+1 = xn − 2(xn + yn)3, yn+1 =1
2yn + (xn + yn)3.
46. Show that one possible second order normal form of the vector field
x = y +O(x2 + y2), y = O(x2 + y2) (33)
isx = y + a1x
2 +O(3), y = a2x2 +O(3) (34)
by completing the following outline:
(i) Write z = x+iy, z = x−iy. Show that in terms of these variables, equation (33) becomes
z =1
2i(z − z) +O(|z|2), (35)
with the equation for ˙z being its complex conjugate.
22
(ii) Show that the homological equation in this case reads
(iii) Let Hn = spanzkzn−k | k = 0, . . . , n. Compute the images of the basis vectors of Hn
under the map L.
(iv) Let n = 2 in part (iii). Show that an appropriately chosen subspace of H2 complementaryto L(H2) is spanned by the polynomial (z + z)2. Conclude that to O(|z|2), the normal formof equation (35) becomes
z =1
2i(z − z) +
a
4(z + z)2 +O(|z|3),
where a = a1 + ia2. Rewriting this equation in terms of the real components x, y, a1, anda2, you should obtain equation (34).
47. Consider the forced van der Pol equation
x+ε
ω(x2 − 1)x+ x = εF cosωt, (36)
with ε 1, 1− ω2 = εσ, and σ = O(1).
(i) Transform (36) into the new set of coordinates (u, v) by the formula
x = u cosωt+ v sinωt, x = −ωu sinωt+ ωv cosωt.
Show that the system of equations for u and v is in the form suitable for applying theaveraging method.
(ii) Average the u− v equations to obtain
u =ε
2ω
[u− σv − u
4
(u2 + v2
)], v =
ε
2ω
[σu+ v − v
4
(u2 + v2
)− F
].
(iii) Rescale the averaged equations as follows:
t→ 2ω
εt, v → 2v, u→ 2u,
and let γ = F/2 to obtain
u = u− σv − u(u2 + v2
), v = σu+ v − v
(u2 + v2
)− γ. (37)
23
0.5
0.5
1.0
1.0
I
II
III
IV
A
B
SN
C
D
E
O
γ
σ
HB
Figure 1: The bifurcation diagram for system (1). Some of the significant points are
A:(σ, γ) =(
1√3,√
827
), O:(σ, γ) =
(12, 1
2
), and C:(σ, γ) = (0, 0).
(iv) Transform (37) to polar coordinates, and show that it undergoes a saddle-node bifurca-tion on the curve
γ4
4− γ2
27(1 + 9σ2) +
σ2
27(1 + σ2)2 = 0.
This is the curve DAC marked SN in Figure 1.
(v) Show that (37) undergoes a Hopf bifurcation on
8γ2 = 4σ2 + 1, |σ| > 1
2.
This is the curve OE marked HB in the Figure 1.
(vi) Show that (37) has a single equilibrium in regions I and III, a sink in I and a source inIII. Show that in region II there are two sinks and a saddle, and in region IV there is a sink,a saddle and a source.
(vii) In Figure 1, consider the broken lines −−− crossing the curves OA, OD, AB, BE, andOB. Draw phase portraits representing the flow on and to each side of the indicated curve.In particular, in what regions does (37) have a limit cycle, and what is its stability type?Help yourself with the fact that at large distances from the origin, (u2 + v2) < 0.
(viii) Are there any inconsistencies in this bifurcation picture, provided no bifurcations otherthan the ones listed occur?
24
(ix) What do the equilibria of the averaged system (37) and their bifurcations signify for theoriginal forced van der Pol equation (36).
48. Consider the mapping P of the unit square into the plane shown schematically in thefigure. Assume that this mapping is a diffeomorphism, and describe its invariant set ΛP andthe orbit structure of P on ΛP . In particular, show that P is topologically conjugate to ashift on three symbols.
49. Consider a two-sided homoclinic tangle such as the one shown in the figure. Describehow you can construct a Smale horseshoe map which is topologically conjugate to the shifton two symbols in such a way that the two symbols denote passages around the left-hand andright-hand portions of the tangle, respectively. In particular, make this statement preciseand prove it.
50. Consider the equation describing the driven and damped pendulum
x+ εδx+ sinx = ε sinωt, (38)
where δ ≥ 0 and 0 ≤ ε 1.
(i) Write equation (38) as a first order system. Show that the extended phase space of thissystem is the cartesian product of a torus and a real line. Also show that, for δ = 0, thissystem is Hamiltonian and find its Hamiltonian function.
(ii) For ε = 0, (38) is a two-dimensional Hamiltonian system. Use this fact to plot its phaseportrait, and find explicit solutions on its two separatrices. What is the physical meaning ofthe motion on these two separatrices?
25
(iii) Use the Melnikov method to compute that, for small ε > 0 and small enough δ ≥ 0,there are two symmetric homoclinic tangles in the phase space of equation (38). State andprove this result as precisely as you can.
(iv) Describe the physical consequences of the two homoclinic tangles that you found in part(iii) by means of an appropriately chosen Smale horseshoe map. Just what does the term“chaotic dynamics” mean in this case?
(v) Do you think that any of the “chaotic” orbits that you have found are stable and canthus be observed experimentally?
51. Consider the Poincare return mapping P : Π0 → Π0, with Π0 being the appropriateportion of the y = 0 plane associated with the Shilnikov phenomenon discussed in class.If the vector field contains a parameter µ whose value is zero exactly when the Shilnikovsaddle-focus connection exists, convince yourself that the mapping P is given by the formula(
xz
)7→
(x(zε
) ρλ[a cos
(ωλ
log zε
)+ b sin
(ωλ
log zε
)]+ eµ+ x
x(zε
) ρλ[c cos
(ωλ
log zε
)+ d sin
(ωλ
log zε
)]+ fµ
)(39)
for some appropriate constants a, . . . , f and x.
(a) Show that under an appropriate rescaling of the variables and parameters, the map (39)becomes (
xz
)7→(αxzδ cos (ξ log z + φ1) + eµ+ xβxzδ cos (ξ log z + φ2) + µ
)(40)
(b) Show that, if we assume |αzδ| 1, solving for the fixed points of the mapping (40)amounts to solving the equation
z − µ = (eµ+ x) βzδ cos (ξ log z + φ2) (41)
(c) Investigate the solutions of equation (41) graphically and show the following facts: If0 < δ < 1, then there are finitely many fixed points for both µ > 0 and µ < 0, andinfinitely many for µ = 0, and if δ > 1, then there are no fixed points for µ ≤ 0 except forthe homoclinic orbit itself at z = µ = 0, and there is precisely one fixed point for µ > 0.Graph the dependence of the periods of the periodic orbits in the x − y − z phase spacethat correspond to these fixed points of the map P on the value of the parameter µ. Notethat the closer z is to z = 0 at one of these fixed points, the longer the period, since z = 0corresponds to a homoclinic orbit.
(d) Find the stability type of the fixed points of the Poincare return map (40) that you foundin part (c). Distinguish the cases δ > 1/2 and δ < 1/2.
26
(e) When the Shilnikov homoclinic orbit is broken, the unstable manifold of the originintersects the section Π0 at the point (eµ+ x, µ). Convince yourself that if the z componentof the image of this point is zero, you will obtain a new homoclinic orbit which passes oncethrough a neighborhood of the origin before falling back into the origin. Show that thiscondition is given by the equation
(eµ+ x) βµδ cos (ξ log µ+ φ2) + µ = 0. (42)
Show, therefore, that there are infinitely many values of the parameter µ for which suchdouble-pulse orbits exist.
HINT: You may want to help yourself by looking at the papers by
P. Gaspard [1983]. Generation of a countable set of homoclinic flows through bifurcation,Phys. Lett. A 97, 1–4.
P. Glendinning and C. Sparrow [1984]. Local and global behavior near homoclinic orbits, J.Stat. Phys. 35, 645–696.
52. Let
u(x) =
0, |x| > ξ,
−U, |x| < ξ,
where U and ξ are positive constants.
(a) Consider the equation
−d2f
dx2+ u(x)f = k2f, −∞ < x <∞.
Compute the functions f = φ(x, k) and f = ψ(x, k), with the asymptotic behavior φ(x →−∞, k) → e−ikx and ψ(x → ∞, k) → e−ikx, that we have discussed in class. Find thecoefficients a(k) and b(k) in the expansion
φ(x, k) = a(k)ψ(x, k) + b(k)ψ(x, k).
Compute the reflection coefficient r(k). Verify explicitly that a(k) is analytic in the half-plane Im k > 0. Find its zeros iκn, and the corresponding eigenfunctions φ(x, iκn).From the formula
φ(x→ −∞, iκn)→ bne−κnx
find the coefficients bn.
(b) Reconstruct the potential u(x) from the spectral data r(k), κn, and bn.
27
53. A linear triatomic molecule is simulated by a configuration of masses and ideal springsthat looks like the following diagram:
Mm mk k
The equilibrium length of the springs is b. Find the eigenfrequencies and normal modes forlongitudinal vibration. Describe the physical meaning of the modes.
54. (a) Show that the differential equation
F
(dnw
dzn,dn−1w
dzn−1, . . . ,
dw
dz, w
)= 0
(note that z does not appear explicitly) can be reduced to an (n − 1)th order equation bytaking u = dw/dz as a function of w.
(b) Find the solution of the equation
(2− w)d2w
dz2−(dw
dz
)2
= w − 2w2
in terms of a function defined by an integral. That is, z =∫g(w) dw, where g(w) is a known
function.
55. If an n-th order differential equation for w(z) is invariant under the transformation
z = λαZ, w = λβW
for all values of the parameter λ and for certain constants α and β, show that it can bereduced to an (n− 1)th order equation for u(ζ), where u and ζ are new variables defined by
(i) w = zγζ,dw
dz= zγ−1u, γ =
β
α, if α 6= 0,
(ii) z = ζ,dw
dz= wu, if α = 0.
28
Use
z3w′′ + 2zww′ − 2w2 = 0,
z2w′′ + z2w′2 + zw′ + w = 0,
w′′ + p(z)w′ + q(z)w = 0,
w′′ + ww′ + zw4 = 0,
w′′ =w3/2
z1/2,
as illustrative examples (and if necessary to stimulate general proofs!).
![Schneider Conzerv Em 6400[1]](https://static.documents.pub/doc/80x56/577d20781a28ab4e1e92f775/schneider-conzerv-em-64001.jpg)
