Hey guys, first post. Looking forward to not only getting help here but also hopefully helping others in their quest for maths knowledge. I'm trying to do the following question but I am having all sorts of troubles!
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.
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 text book I got from the library).
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!