# ODE problem

1. Feb 5, 2006

### Tony11235

Let x = x1(t), y = y1(t) and x = x2(t), y = y2(t) be any two solutions of the linear nonhomogeneous system.

$$x' = p_{11}(t)x + p_{12}(t)y + g_1(t)$$
$$y' = p_{21}(t)x + p_{22}(t)y + g_2(t)$$

Show that x = x1(t) - x2(t), y = y1(t) - y2(t) is a solution of the corresponding homogeneous sytem.

I am not sure what it is that I am suppose to do. Could anybody explain?

2. Feb 6, 2006

### HallsofIvy

Staff Emeritus
"Plug and chug". The "corresponding homogeneous system" is, of course, just the system with the functions g1(t) and g2(t):
$$x'= p_{11}(t)x+ p_{12}(t)y$$
$$y'= p_{21}(t)x+ p_{22}(t)y$$
replace x with x1- x2, y with y1- y2 in the equations and see what happens. Remember that x1, x2, y1, y2 satisfy the original equations themselves.

Last edited by a moderator: Feb 6, 2006