# Systems of First Order Linear Equations

## Homework Statement

Systems of first order equations can sometimes be transformed into a single equation of
higher order. Consider the system

(1) x1' = -2x1 + x2
(2) x2' = x1 - 2x2

Solve the first equation for x2 and substitute into the second equation, thereby obtaining a second order equation for x1. Solve this equation for x1 and then determine x2 also.

## The Attempt at a Solution

Solving (1) for x2 yields:

x2 = x1' + 2x1

Substituting into (2) yields:

x2' = x1 - 2(x1' + 2x1)

Simplifying...

x2' = -3x1 - 2x1'

How am I supposed to solve this now with the x2' term there?

HallsofIvy
Homework Helper
You aren't. You haven't yet reduced it to a higher order equation in one variable.
"Solve the first equation for x2 and substitute into the second equation, thereby obtaining a second order equation for x1" obviously won't give you a second order differential equation. I think you have dropped the first 2/3 of the instruction!

Start by differentiating the $x_1''= -2x_1+ x_2'$. Now use the second equation to replace that $x_2'$: $x_1''= -2x_1+ (x_1- 2x_2)= -x_1+ 2x_2. NOW solve the first equation for [itex]x_2$ and substitute that into the equation $x_1''= -x_1+ 2x_2$.