# Thread: Systems of ODE, using Laplace Transforms

1. ## Systems of ODE, using Laplace Transforms

Hi everyone, this is my first post here, but hopefully ill be sticking around and having a bit input!

Anyways, im revising for the end of year exams and as usual with maths one of the best ways to get good is just to practice, practice practice! For a module on all sorts of Differential Equations, which wouldn't be complete without Laplace transforms, ive been doing just that and hit a snag on one question which i need help with more for the sake of my own sanity than anything else! I think ive just went about it the wrong way, missed the obvious and ended up in an algebraic swamp!

So say I had to solve

2x' + 3y' +7x = 14t + 7
5x' -3y' + 4x + 6y = 14t -14

given x(0)=y(0)=0 (thankfully!)

So I take the laplace transforms first off (sorry i havent got the hang of latex yet!) (NB in my notation, the lower case x has changed to X like you may see on paper with a line above it! f(t)=F(s) kinda thing)

2sX+3sY+7X = 14/s^2 + 7/s
5sX -3sY +4X +6Y = 14/s^2 -14/s

Rearanging...

X(2s+7) +3sY= s(7s+14)/s^3
X(5s +4) + Y(6-3s)=s(14-14s)/s^3

Using the first one

Y=s(7s+14)/3s^4 - X(2s+7)/3s

And putting it back into the second one.

I get it down to the stage

X= (-21s^2 +42s -84 ) /( s^2 (9s^2 +3s -84))

I cannot think of any tricks or ways of factorising this!!!! Im willing to bet ive made or have forgot something simple to the point of embarrasment but i am meant to be revising but instead have spent 3 hours on a whiteboard trying to do this, and it is making me feel thick!

The idea was to get that, factorise so things cancell out quite nicley, mabey do a partial fraction, then do an inverse transform using a standard transform, then use x to find y!

Can anyone point me in the correct direction or have and ideas.

Again, sorry that it looks more like matlab code than anything else!

2. I didn't work the Algebra myself but here is a maple screenshot of what they should be.

My guess is you have an algebra error

I have found for me elimination works better that substition

try multiplying the top equation by

$\displaystyle (6-3s)$ and the bottom by $\displaystyle -3s$