# Thread: coupled 1st order ODEs, and initial conditions...

1. ## coupled 1st order ODEs, and initial conditions...

Hi, I'm having some trouble with this problem...if anyone could help me out a bit that'd be great - it's really just one bit of it i don't get.

Solve the pair of differential equations:

dx/dt = ax dy/dt = ay + bx

where a and b are arbitrary constants. All subject to the initial conditions x(0)=2 , y(0) = 1.

So basically I converted it to a 2nd order ODE (d2y/dx2= ...) as this is the only method i've been taught, and found the general solution - all seemed to be ok up until then. The general solution was, i think, of the form y = A exp(at) + Bt exp(at). (Where A and B are constants)

From there, i can plug the y(0) condition in, and get an expression which tells me A is 1. But i've no idea how to use the x(0) initial condition, or indeed what it really means (x at t=0?), and i'm pretty certain i need to do so in order to find both A and B...

Any advice would be hugely appreciated! Thanks.

2. Can you try

$
\frac{dy}{dx}= \frac{\frac{dy}{dt}}{\frac{dx}{dt}}= \frac{ay+bx}{ax}
$
$= \frac{ay}{ax}+\frac{bx}{ax}= \frac{y}{x}+\frac{b}{a}$