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Homework Statement
Solve the system of first order differential equations using Laplace Transforms:
dx/dt = x - 4y
dy/dt = x + y,
subject to the initial conditions x(0)=3 and y(0)=-4.
Homework Equations
So far I've used the limited knowledge of Laplace Transforms for first order ODE's to get this far:
L[x`] = L[x] - L[4y]
s*L[x] - x(0) = L[x] - L[4y]
s*L[x] - L[x] - 3 = -L[4y]
(s-1)*L[x] = 3 + L[4y] <--------- Equation 1
L[y'] = L[x] + L[y]
s*L[y] - y(0) = L[x] + L[y]
s*L[y] + 4 = L[x] + L[y]
(s-1)*L[y] = L[x] - 4 <----------Equation 2
or (s-1)*L[y] + 4 = L[x]
Up to this stage I am kind of confident I have been using Laplace Transforms right (from the couple of examples I have in a textbook I got from the library).
The Attempt at a Solution
The step where I become very confused is substituting equations 1 and 2 into one another to evaluate y(t) and x(t).
When I substitute (2) into (1) i get the following:
(s-1)[(s-1)*L[y] + 4] = 3 + L[4y]
From here I have probably tried 20 different ways of getting a solution for y(s) but every single one is very complicated and leads to a dead end for me (they are way too long to type). Because of this I suspect I am doing something wrong here [potentially I am even applying Laplace Transforms completely wrong!].
I am hoping someone knows where I am going wrong or what I'm doing wrong. Any help and advice would be greatly appreciated, thanks!