(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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Laplace Transforms to solve First Order ODE's.

**Physics Forums | Science Articles, Homework Help, Discussion**