How to Solve Coupled Differential Equations?

[mkrems]

Hi all,

I want to solve equations of the form:
\dot x + x + y = sin(\omega t)
\dot y = \dot x - y

This is not a standard type of form for Runge-Kutta or linear systems of equations because
\dot y = f(\dot x, y, t)
instead of
\dot y = f(x, y, t).
Any hints or links to place for help would be appreciated! Thanks!

 

[tiny-tim]

Hi mkrems!

Can't you get it into f(x, y, t) form by substituting for x' from the first equation?

 

[Matthew Rodman]

Use your first equation to isolate y, namely,

y = \sin{\omega t} - x^{\prime} - x

Now, differentiate this to get y prime,

y^{\prime} = \omega \cos{\omega t} - x^{\prime \prime} - x^{\prime}

and substitute these into your second equation to get...

\omega \cos{\omega t} - x^{\prime \prime} - x^{\prime} =<br /> x^{\prime} - \sin{\omega t} + x^{\prime} + x

which may be rearranged to give you a (soluble) second-order equation in x only.

x^{\prime \prime} + 3 x^{\prime} + x = \omega \cos{\omega t} + \sin{\omega t}