Analyzing the Linear Nonhomogeneous System

[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?

 

[HallsofIvy]

"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.