Homework 18
For the following ve problems, a homogeneous second-order linear dierential equation, two functions y1 and y2 and a pair of initial conditions are given.
First verify that y1 and y2 are solutions of the dierential equation. Then, nd
a particular solution of the form y = c1 y1 + c2 y2 that satises the given initial
conditions. Primes denote derivatives with respect to x.
1.
2.
3.
4.
5.
y 00 + y 0 = 0;
y 00 + 25y = 0;
y1 = cos 5x, y2 = sin 5x;
y (0) = 10, y 0 (0) =
10
2, y0 (0) = 8
y 00 2y 0 + 2y = 0; y1 = ex cos x, y2 = ex sin x; y (0) = 0, y 0 (0) = 5
x2 y 00 + 2xy 0 6y = 0; y1 = x2 , y2 = x 3 ; y (2) = 10, y 0 (2) = 15
x2 y 00 + xy 0 + y = 0; y1 = cos(ln x), y2 = sin(ln x); y (1) = 2, y 0 (1) = 3
y1 = 1, y2 = e x ;
y (0) =
The following problem illustrates the fact that the superposition principal
does not generally hold for nonlinear equations.
6. Show that 1=x is a solution of y0 + y2 = 0, but that if c 6= 0 and c 6= 1
then y = c=x is not a solution.
For the following three problems, determine whether the given pairs of functions are linearly independent or linearly dependent on the real line.
7. f (x) = , g(x) = cos2 x + sin2 x
8. f (x) = sin2x, g(x) = 1 cos 2x
9. f (x) = ex sin x, g(x) = ex cos x
For the following three problems, nd the general solution of the given differential equations.
10.
11.
12.
y 00
3y0 + 2y = 0
4y00 + 4y0 + y = 0
35y00 y0 12y = 0
1