Finding an ODE given a vector-matrix

[tracedinair]

Homework Statement

Let x'=Ax, where A =

|1 2|
|3 4| and x =

|x₁(t)|
|x₂(t)|

Find a scalar second order linear homogeneous ODE, with x₁(t) as the dependent variable, which arises by the elimination of the variable x₂(t) from the vector matrix equation.

Homework Equations

The Attempt at a Solution

I'm not really sure where to begin. My guess is to write out two scalar equations to start with but after that I do not know where to go.

 

[lippyka]

Any help would be appreciated. Let's take the two equations given by the vector matrix equation: x1' = 1*x1 + 2*x2 x2' = 3*x1 + 4*x2 If we eliminate x2 from the equations, we obtain: x1' = 1*x1 + (3/2)*x1' Substituting x1' for x1, we obtain: x1" = (3/2)*x1' Therefore, the scalar second order linear homogeneous ODE with x1(t) as the dependent variable is x1" = (3/2)*x1'.