1. ## Solving differential equations

Hi

I am not very advanced in calculas.

I need to solve for X(t)/Y(t) when t -> infinity.

X'(t)+aX(t)+cX(t)-bY(t)=0 ....1 X(0)=0
Y'(t)+bY(t)+cX(t)=0 .....2 Y(0)=0

So I thought of deriving both equations to get:

X''(t)+aX'(t)+cX'(t)-bY'(t)=0 ...3
Y''(t)+bY'(t)+cX'(t)=0 ...4

Then substituting 2 into 3 & substituting 1 into 4

X''(t)+aX'(t)+cX'(t)-b*[bY(t)]-bcX(t)=0 ....5 Y''(t)+bY'(t)+cX(t)[a+c]+bY(t)=0 .....6

Then substituting 1 into 5 Then substituting 2 into 6

X''(t)+aX'(t)+cX'(t)+bX'(t)+baX(t)=0 Y''(t)+[a+b+c]Y'(t)+[ab+cb+b]Y(t)=0

Taking the Laplace transform Taking the Laplace transform

X(s)[s^2+s+a+c+sa+sc+sb+ba]=0 Y(s)[s^2+sa+2sb+sc+sab+scb-s-b]

Then

X(s)/Y(s)=[s^2+sa+2sb+sc+sab+scb-s-b]/[s^2+s+a+c+sa+sc+sb+ba]

This is where I get stuck as I don't know how to transform it back to get X(t)/Y(t)

Any help would be much appreciated.

Thank you

2. ## Re: Solving differential equations

Originally Posted by Mildred
Hi

I am not very advanced in calculas.

I need to solve for $\frac{x(t)}{y(t)}$ when $t \rightarrow\infty$

$\begin{cases} x'(t)+ax(t)+cx(t)-by(t)=0\\ y'(t)+by(t)+cx(t)=0 \end{cases}$

Given $x(0)=0$ and $y(0)=0$

So I thought of deriving both equations to get:

$\begin{cases} x''(t)+ax'(t)+cx'(t)-by'(t)=0\\ y''(t)+by'(t)+cx'(t)=0 \end{cases}$
It would seem ill advised to me to make this a second order system considering you're only given two initial conditions(ICs). Basically anytime you have a DE when you solve them you're always doing some type of integration in order to solve them and in order to fully solve a DE you need the same number of initial conditions as the highest order derivative in the DE. In this case you are give 2 first order equations with 2 initial conditions so the system is solvable; however, in your first step you turn the system into a second order system of 2 equations so now you would need 4 initial conditions in order to solve it. Since you're not given 4 ICs you should keep this a first order system.

You're on the right track as far as taking the Laplace Transform of both equations. Take the Laplace Transform of the above system and solve it for $x(t)$ and $y(t)$.