(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant 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 text book I got from the library).

3. 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!

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# Homework Help: Laplace Transforms to solve First Order ODE's.

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