Assignment 2
Due date: July 21, 2015
For Problem 1, 2, 3, 4, 5 and 6, you only need to three of them.
1. For the differential equation 2x2 y 00 + (3x − x2 )y 0 − y = 0 for x > 0, find the first three terms of a non-zero
series solution at x = 0, satisfying lim y(x) = 0.
x&0
2. Consider the differential equation 2(x − 1)y 00 + y 0 + y = 0 for x > 1.
(a) Show that x = 1 is a regular singular point. Find the indicial equation and the exponents at x = 1.
(b) Find the first two non-zero terms of two linearly independent series solutions about x = 1.
(c) Find the first two non-zero terms of the series solution about x = 1, satisfying lim y(x) = 1 and
x&1
√
lim x − 1y 0 (x) = 1.
x&1
3. Consider the Laguerre differential equation:
xy 00 + (1 − x)y 0 + λy = 0,
x > 0,
where λ is a real constant.
(a) Show that x = 0 is a regular singular point. Find the indicial equation and the exponents at x = 0.
(b) Find the general series solutions near x = 0.
(c) Show that if λ is a non-negative integer n, then there is a polynomial solution of degree n.
(d) Find a polynomial solution for each of the case λ = n = 0, 1, 2, 3.
4. Consider the differential equation:
x(x − 1)y 00 + 6x2 y 0 + 3y = 0,
x > 0,
(a) Show that x = 0 is a regular singular point. Find the indicial equation and the exponents at x = 0.
(b) Find the general series solutions near x = 0. What would you expect the radius of convergence of the
series solution be?
9
2 00
0
2
5. Find the general series solution near x = 0 to x y + xy + x −
y = 0.
4
6. Consider the differential equation:
x3 y 00 + αxy 0 + βy = 0,
where α and β are real constants and α 6= 0.
(a) Show that x = 0 is an irregular singular point.
∞
X
(b) Find a solution of the from
an xr+n with a0 = 1.
n=0
x > 0,
7. Solve the transport equation with the initial condition:
ut + 3ux = 0, x ∈ R, t > 0.
u(x, 0) = cos(x), x ∈ R.
8. Solve the transport equation with the initial condition:
ut + 2ux = ex+t , x ∈ R, t > 0.
u(x, 0) = sin(x), x ∈ R.
9. Determine whether the method of separation of variables can be used to replace the given PDEs by a pair of
ordinary differential equation. If so, find the equations.
(a) xuxx + ut = 0.
(b) uxx + uxt + ut = 0.
(c) uxx + (x + y)uyy = 0.
(d) [p(x)ux ]x − r(x)utt = 0.
(e) uxx + uyy + xu = 0.
10. Solve y 00 + y = 0, y(0) = 0 and y 0 (π) = 1 if the solution exists.
11. Solve y 00 + y = 0, y 0 (0) = 1 and y(π) = 0 if the solution exists.
12. Use separation of variables to find

PDE:



BC:



IC:
13. Use separation of variables

PDE:




BC:



 IC:
the solution to the heat conduction problem:
100uxx = ut ,
0 < x < 1, t > 0,
u(0, t) = 0, u(1, t) = 0,
t > 0,
u(x, 0) = sin(2πx) − sin(5πx),
0 ≤ x ≤ 1.
to find the solution to the heat conduction problem:
uxx = 4ut + u,
0 < x < 2, t > 0,
u(0, t) = 0, u(2, t) = 0,
t > 0,
πx − sin(πx) + 4 sin(2πx),
u(x, 0) = 2 sin
2
Page 2
0 ≤ x ≤ 2.