MATH2221 Higher Theory and Applications of Differential Equations 2021
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MATH2221
Higher Theory and Applications of Differential Equations
2021
1. i) Find the general solution of the following equation. x y2\\ . 2xy + 2y\ = x3
ii) Use variation of parameters (NO credit for any other method) to solve
the follow equation.
y\\ . 2y\ . 3y = e 
.
iii) Consider the following initial value problem.
y\\ . 9y = 0, y(0) = .1, y\ (0) = 2
a) Solve the differential equation using the power series method.
b) Directly solve the differential equation (i.e., using the characteristic equation) and show that the result is consistent with part (a).
2. i) Put
A := 
┐
and consider the linear system
北\ (t) = A北(t).
a) Find the eigenvalues and eigenvectors of A.
b) Use the result of part (a) to write the general solution of the system.
c) Suppose that eAt = Bc0 for some 2 x 2 matrix B and constant vector c0 e R2 . Directly from your answer to part (b) identify the matrix B .
d) Based on your answer to part (b) explain the nature of the stability of the equilibrium solution 北eq (t) = 0.
e) Use your answer to part (b) to sketch the phase portrait for the system. Include some representative solution trajectories as well as the “eigendirections” .
ii) Consider the following differential operator L. Lu := u\\ . 3u\ + 2u.
a) Determine the solution to the homogeneous operator equation
Lu = 0.
b) Consider the inhomogeneous operator equation
Lu = e5北 .
What operator annihilates e5北?
c) Use your answer to part (b) to help you find the general solution of Lu = e5北 .
d) Explain how your answer to part (d) changes if the inhomogeneous equation is changed to Lu = . 
e5北 .
e) Write the operator Lu in its formally self-adjoint form.
3. i) Consider the following ODE:
╱ 1 . x2、y\\ . 2xy\ + λy = 0, . 
< x < 
.
a) Transform (1) into Sturm-Liouville form.
b) If the eigenfunctions of (1) were to form an orthogonal set on the in- terval ┌ . 
, 
┐ , what equality must any two of these functions obey?
ii) Consider the function f (x) = x2 on (.π, π). You may assume that f has the Fourier series representation on (.π, π) given by
x2 = 
+ 4 - 
cos (nx),
which is generated by the complete orthogonal system
{1, cos x, cos (2x), cos (3x), . . . }, . π s x s π .
a) Use the above facts together with Parseval’s identity to prove that
π4 - 1 1 1 1 1
90 j4 24 34 44 54
b) Consider the function
cos (nx)
2 .
Use Parseval’s identity together with the result of part (a) to find
the exact value of m
╱g(x)、2 dx.
_m
iii) Consider the following partial differential operator for a function u :
R3 → R.
Lu := .3u年年 + 5u年』 . 7u之之 .
Is L elliptic? Justify your answer.
4. Consider the temperature u(x, t) in a bar of length π . Assume that the tem-
perature distribution u satisfies the one-dimensional diffusion PDE, namely ut (x, t) = u北北 (x, t),
at position x e (0, π) and time t > 0. The left end of the bar is held fixed at a temperature of 0, whereas the right end is insulated so that no heat can
escape through it. Thus, the bar satisfies the boundary conditions
u(0, t) = u北 (π, t) = 0, t > 0.
i) Using the method of separation of variables let u(x, t) := X(x)T (t) and derive the following ordinary differential equations,
T\ . kT = 0, X\\ . kX = 0,
where k is the separation constant.
ii) Consider the three cases k = 0, k < 0 and k > 0 and solve for X(x).
iii) Find all possible solutions Tn (t) for T (t).
iv) The initial temperature distribution is
u(x,0) = 500, 0 s x s π .
Use this to find a formula for the temperature u(x, t). As part of this
question you may use without proof the fact that {sin ╱ 
、} 
forms a complete orthogonal collection on [0, π] and, furthermore, that
0 m sin2 ╱ 
x、 dx = 
π , n e … .
v) Find
lim u(x, t).
t→-
Does the answer seem physically plausible given the boundary data? Ex- plain briefly. (Note: To answer this question you may assume that you can interchange the limit and the summation.)
2022-08-16