Differential equations, comprehensive exam topics and sample questions 1 Topics covered

Differential equations, comprehensive exam
topics and sample questions
1
Topics covered
• ODE’s: Chapters 1-5, 7, from Elementary Differential Equations by Edwards and
Penney, 6th edition.
1. Exact solutions of first and second order ODEs: separable, linear, exact, integrating factor, of form
y 00 = F (y, y 0 ) and y 00 = F (x, y 0 ), linear n-th order, variation of parameters.
2. Series solutions: various singularities, method of Frobenius.
3. Laplace transform methods: Table will be given (see attached). Transforms for periodic, functions,
step function, and delta functions.
4. Linear systems: exact solutions; fundamental matrix; inhomogeneous systems; stability; phase
portraits, classification of equilibria.
5. Stability and phase analysis: equilibria, stability, phase portrait, Hopf bifurcation.
• PDE’s: Chapters 8,9 from Edwards&Penney; Chapters 1,2,4,5,6, 7, 14.1 from Partial
Differential Equations by Walter Strauss
1. Derivation of diffusion/heat equation and wave equation
2. Maximum principle
3. Heat and wave equation on entire line; heat kernel
4. Separation of Variables, Strurm-Liouville problems, Eigenfunction Expansions (Chap. 8,9 of E-P
or chap 4-5 of Strauss)
5. Method of Characteristics, including nonlinear PDE’s and shock formation (chap.
Strauss)
6. Fourier transform, Laplace transform.
• Vector calculus: see Adams, multivariable calculus.
1. Divergence, Stokes, Gauss theorems; computing fluxes.
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Some sample exam questions
1. Find the general solution to each one the following ODE’s:
(a) y 00 − 6y 0 + 9y = exp(−3x)
(b) y 00 + y = tanh(x)
(c)
y 00
y0
1
−
=−
0
y
y
t
y2 − x
2xy
00
(e) y = yy 0
(d) y 0 =
1
14.1 from

1
2. Let M =  0
0
1
1
0

0
0 .
−1
(a) Compute exp (tM ) .
(b) Determine the solution to y˙ = M y subject to y(0) = (0, 2, 0) .
(c) Determine the general solution to


0
y 0 = M y +  et  ; y(0) = (0, 2, 0).
0
(d) Solve the following initial value problem using the Laplace transform,
y 00 + 4y = δ (t − π) + δ(t − 2π) + δ (t − 3π) ,
y 0 (0) = 0.
y (0) = 0,
Here δ is the Dirac delta function. Sketch the resulting solution.
(e) Use Laplace transform to find the solution to
tx00 (t) + 2x0 (t) + tx = 0;
x(0) = 1, x0 (0) = 0.
(f) Use Laplace transform and the convolution to show that the solution to
x00 + 2x0 + x = f (t); x(0) = 0 = x0 (0)
is given by
Z
x(t) =
t
τ e−τ f (t − τ )dτ.
0
3. Consider the Hermite ODE,
u00 − 2xu0 + λu = 0,
−∞ < x < ∞.
(a) Compute the recursion relation for the power series solution u =
(1)
P∞
0
an xn .
(b) Show that (1) has a polynomial solution of degree n whenever λ = 2n. Denote such a solution
by Hn (x) .
R∞
(c) Show that there exists a function w (x) > 0 such that −∞ Hn (x) Hm (x) w(x) = 0 whenever
n 6= m. Find w (x) .
4. Consider the ODE system modelling a certain two-species interaction,
0
x = x (1 − (x + y))
, x, y ≥ 0.
y 0 = y (2 − (x + y))
Here, x and y represent two competing species.
a) Plot the nullclines of the system and find the equilibria points.
b) Classify the stability of each of the equilibria and sketch the phase portrait of the system.
c) What is the eventual state of the system, in the limit t → ∞, if initially x > 0 and y > 0?
5. Consider the system
x˙ = −y + µx + xy 2 ,
y˙ = x + µy − x2 .
a. Analyse the stability of the zero equilibrium and classify any bifurcations that occur as µ is
varied.
b. Find all equilibria when µ = 0 and classify their stability.
c. With µ = 0, sketch a phase plot diagram, labelling nullclines and equilibrium points.
2
6.
(a) Let u(x, t) be the vertical displacement of an elastic homogenous string undergoing small
transverse vibrations. Carefully derive the wave equation ρutt = T uxx , where T is the tension
magnitude and ρ is its linear density.
(b) In the derivation above, you’ve neglected the gravity. How does the equation change if you
incorporate the gravity acting in the vertical direction?
7.
(a) Carefully derive the one-dimensional diffusion equation.
(b) Suppose that a pollutant is diffusing in a river. The river moves at speed v from left to right.
Derive the PDE describing this situation.
8. Solve the Wave equation utt = uxx on the entire plane subject to the initial conditions u (x, 0) =
φ(x), ut (x, 0) = 0 where φ(x) is given by
1, if |x| < 1
.
φ(x) =
0, if |x| ≥ 1
Sketch your solution for t = 0, 0.5, 1 and 1.5.
9.
(a) State the maximum principle for the Heat equation.
(b) Consider the heat equation with drift: ut = uxx + vux where v is some constant. Does the
maximum principle also hold for this PDE? Explain why or why not.
(c) Does the maximum principle still hold for the heat equation ut = uxx if “=” is replaced by
“≥”? by “≤”?
10.
(a) What is the general solution to the Wave equation utt = uxx ? What is its solution when the
u(x, 0) = φ(x) and ut (x, 0) = ψ(x)?
(b) Consider the PDE 3utt + uxx − 4uxt = 0. Is it elliptic, parabolic or hyperbolic? Find its general
2
solution. HINT: 3t2 + x2 − 4xt = (x − 2t) − t2 .
11. Show that the solution to the problem ut = uxx + f (x, t), u(x, 0) = φ(x), u(0, t) = h(t), u(1, t) =
g(t) is unique.
12. A rod is is heated from the right end and cooled at the left end. Initially, the temperature is zero.
This situation can be modelled as:
ut = uxx ;
u(0, t) = 0,
u(x, 0) = 0.
u(1, t) = 1
(a) Determine u(x, t).
(b) Let v(x) = u(x, ∞) = limt→∞ u(x, t). Determine v(x).
(c) Show that u(x, t) ≤ v(x) for all t > 0, x ∈ (0, 1)
3
13. Let D = (x, y) : y > 0 and x2 + y 2 < 1 , be an upper half-disk and suppose that
∆u = 0 inside D
and moreover u has the following boundary conditions:
u = 0 on the right-bottom boundary y = 0 and x > 0;
∂n u = 0 on the left-bottom boundary y = 0 and x < 0;
u = 1 on the top boundary (x, y) = (cos θ, sin θ).
Find the solution for u in terms of an appropriate infinite series.
14. Consider the system
ut + (2 − u)ux = 0;
u(x, 0) = 1 + tanh(x).
(a) Determine the characteristic curves for this ODE.
(b) Show that the solution develops a shock and compute the time t = ts at which the shock first
occurs.
(c) Sketch the solution profile u(x, t) for t = 0, 0.5, 1, 1.5.
15. Consider the PDE ut − uux = 0 subject to the initial condition u(x, 0) = φ(x) where φ(x) is a
”tent” function given by
1 − |x| , |x| < 1
φ(x) =
.
0,
|x| ≥ 1
(a) Solve for the characteristics and sketch them.
(b) Sketch the solution for t = 0, 0.5, 1, 1.5.
(c) Does the solution exhibit a shock? If yes, determine tshock , the time when the shock first
occurs.
16. Solve the PDE ut + 2xux = t subject to the condition u(x, 0) = x2 .
17. The temperature u(x, t) in a rod of length L satisfies ut = uxx with boundary conditions ux (0, t) =
1, u (L, t) = 0 and initial conditions u (x, 0) = x.
a) What is the steady state solution?
b) Determine an expansion for u (x, t) in terms of appropriate eigenfunctions. Evaluate all coefficients explicitly.
18. Let λ1 < λ2 < · · · be the eigenvalues and y1 , y2 , . . . the corresponding eigenfunctions of the following
boundary value problem on [0, 1]:
y 00 + λy = 0,
y (0) = 0 = 2y (1) + y 0 (1)
a) Derive an implicit formula for λ. Give an approximate formula for λn for large n.
b) Obtain a formula for the eigenfunctions yn . What orthogonality relation do they satisfy?
c) Consider the boundary value problem u00 + µu = f (x)P
with u (0) = 0 = 2u (1) + u0 (1) , µ > 0.
Suppose that f can be expanded in terms of yn (x) as f = cn yn (x) for some coefficients cn . Derive
the solution to this problem in terms of the eigenfunctions yn .
19. Solve the following problem for the Laplace’s equation on the exterior of the unit disk:

 urr + 1r ur + r12 uθθ = 0, 0 < θ ≤ 2π, r > 1
ur (1, θ) = 2 cos θ − 3 sin 3θ

u (r, θ) is bounded for r ≥ 1.
4
20. The telegraph equation is given by
∂2u
∂2u
∂u
=
+ 2b
+ au .
2
2
∂x
∂t
∂t
u(x, 0) = f (x) ,
∂u
(x, 0) = g(x) ,
∂t
−∞ < x < ∞ , t > 0 .
c2
(a) Set U(ω, t) =
1
2π
R∞
−∞
u(x, t)eiωx dx and solve for U.
(b) In the special case where b2 = a solve for u(x, t).
You may find the following Fourier transforms useful for this problem
1 iωa
= F(δ(x − a)) ,
e
2π
1 sin(aω)
= F(v(x)) ,
π
ω
cos(ωa)
= F((δ(x + a) + δ(x − a)))
π
Where,
(
1
v(x) =
0
if |x| ≤ a
if |x| ≥ a
Z ∞
1
f (x)eiωx dx
F(f (x)) =
2π −∞
Z ∞
L(f (t)) =
f (t)est dt
0
21. Let Ω be the top half of unit disk, Ω = {(r cos θ, r sin θ) : 0 < r < 1, θ ∈ (0, π)}. Consider the
following Green’s function:

∆G = δ(¯
x−x
¯0 ) inside Ω

∂n G = 0 on bottom boundary of Ω (y = 0)

G = 0 on top (circle) boundary of Ω (r = 1)
(a) Write down an explicit formula for G. (hint: try the method of images)
(b) Solve the following problem in terms of the Green’s function you just found:

∆u = 0 inside Ω

∂n u = f (x) on bottom boundary of Ω (y = 0)

u = g(θ) on top (circle) boundary of Ω (r = 1)
Simplify your final answer as much as you can.
22.
(a) Suppose that ∆u = 0 for all x ∈ R2 . Show that for any smooth domain D, and for any x0 ∈ D,
we have:
Z
1
u(x0 ) =
{∂n ln |x − x0 | u(x) − ln |x − x0 | ∂n u(x)} dS(x).
2π ∂D
5
(b) Deduce from part (a) that
u(0, 0) =
1
2π
Z
2π
u(cos(θ), sin (θ))dθ.
0
23. Consider the PDE ut = uxx subject to initial condition u(x, 0) = cos(x). By solving this PDE in
R∞
2
two distinct ways, find the value of −∞ e−ay cos(y)dy with a > 0.
24.
(a) Briefly state Stokes, Green and Divergence theorems.
(b) A smooth surface S lies above the plane z = 0 and has as its boundary the circle x2 + y 2 = 4y
in the plane z = 0. This circle is also the boundary of a disk D in that plane. The volume of
the 3-dimensional region R bounded by S and D is 10 cubic units. Find the flux of
F (x, y, z) = (x + x2 y)i + (y − xy 2 )j + (z + 2x + 3y)k
through S in the direction outward from R.
2
z2
(c) Let
R F = 2xzi + e j+ x − x cos (y) k. Using one of the three theorems stated in Part a, find
F · dr where c is the boundary of the unit sphere intersected with the xz plane, traversed
c
in the couterclockwise direction when viewed from (0, 1, 0) .
(d) Find the flux of F = xy 2 i + x2 yj + k outward through the hemispherical surface x2 + y 2 + z 2 =
4, z ≥ 0. Hint: You can use the Divergence theorem to simplify the computation.
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Table of Laplace Transforms (will be given during the exam)
f (t) = L−1 {F (s)}
F (s) = L{f (t)}
1
1
, s>0
s
eat
1
, s>a
s−a
tn n positive integer
n!
, s>0
sn+1
tp , p > −1
Γ(p + 1)
, s>0
sp+1
sin(at)
a
, s>0
s2 + a2
cos(at)
s
, s>0
s2 + a2
tn eat , n positive integer
n!
, s>a
(s − a)n+1
H(t)1
e−cs
, s>0
s
H(t)f (t − c)1
e−cs F (s)
ect f (t)
F (s − c)
f (ct)
1 s
F
, c>0
c
c
Rt
0
1 H(t)
f (t − τ )g(τ ) dτ
F (s)G(s)
δ(t − c)
e−cs
f (n) (t)
sn F (s) − sn−1 f (0) − · · · f (n−1) (0)
is the Heaviside function
7