Can you solve this?

Question

Solve x'=(5, 3, -1, 1)x, x(0)=(2, -1).

 

I know that the eigenvalues are 2, 4. And the eigenvector for 2 is (1, 3) and for 4 is (1, 1). But how do I find the numbers in front of these 2 eigenvectors? (5, 3, is on the left, -1, and 1 are on the right.) Please help.

Answer 1

This is an initial value problem (IVP). You found the eigenvalues and corresponding eigenvectors, so you know the general solution is

 

x(t) = c₁[1,3]e^2t + c₂[1,1]e^4t

 

To fix the two constants c₁ and c₂, you need to impose the initial condition, x(0)=[2,-1].

 

You get

 

x(0)=[2,-1]=c₁[1,3] + c₂[1,1]

 

This you can write as a system of 2 equations:

 

2=c₁+c₂

-1=3c₁+c₂

 

You could solve this by putting it into matrix form again, but it's just as easy to solve it "by hand": subtract the second from the first equation and get

 

3=-2c₁, so that c₁=-3/2, which then implies c₂=7/2.

 

So the solution to the IVP is

 

x(t) = -3/2[1,3]e^2t + 7/2[1,1]e^4t