Systems of First Order Linear Equations

[capertiller]

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.

Homework Equations

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]

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 $x_2'$: [itex]x_1''= -2x_1+ (x_1- 2x_2)= -x_1+ 2x_2.

NOW solve the first equation for $x_2$ and substitute that into the equation $x_1''= -x_1+ 2x_2$.