We are learning how to solve systems of linear ODE's by elimination. I am not sure how to solve this:

Question

x'=-y+t
y'=x-t

I took the derivative of the second eqn to get
y''-x'=-t'
and added the two to get
y''+y=-t'+t
but I don't think thats the right way to proceed. Suggestions?
tia

Answer 1

x'=-y+t
y'=x-t

Add these two equations, you get:

x' + y' = x - y

y' + (x')y = x

That gives you a linear differential equation of the form:

y' + Py = Q

Considering that P is the derivative of x, it'll make things easy while solving for the integrating factor.

Answer 2

The equation should be:

x' + y' = x - y
=> y' + (1)y = (x - x')