Laplace Transform of Systems of ODEs with variable coefficients

[physics19921]

Homework Statement

Say you have:

EQ1: y1''*t+y1'*t+y2=0

and

EQ2: y2''*t+y2'*t+y1=0

y1(0)=0,y1'(0)=0,y2(0)=0,y2'(0)=0

Homework Equations

The Attempt at a Solution

I can get it so far, but having both y1 and y2 really gives me fits:

Eq1: Y1(-2s-1)+dY1/ds(-s²-s)=-Y2

Eq2: Y2(-2s-1)+dY2/ds(-s²-s)=-Y1

I try to shift it around to be dY1/Y1 = (-2s-1)/(-s²-s)-Y2/Y1(-s²-s)

But then I just don't know to do the integrations given you have both Ys. I just can't separate them.

Any help would be appreciated.

 

[I like Serena]

Welcome to PF, physics19921!

You can't separate the Ys?
So don't.
Add them, and subtract them respectively.

That is, add the 2 equations and solve for z1=(y1+y2).
Then subtract the 2 equations and solve for z2=(y1-y2).
Finally combine the solutions...