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...

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

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

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