How Do You Solve These Simultaneous Differential Equations?

[masej]

Homework Statement

I need to solve these two equations to obtain expressions for $X(t)$ and $Y(t)$

Homework Equations

[1]. $$\frac{d(X(t))}{dt} = Q \cdot Y(t)$$

[2]. $$\frac{d(Y(t))}{dt} = -Q \cdot X(t)$$

The Attempt at a Solution

Perhaps rearrange equation [1] to get in terms of $Y(t)$ then input this into expression into equation [2] to get equation ([3]) just with $X(t)$ terms. Then solve [3] to find $X(t)$ and input this back into equation [1].

.. hows that? It gets rather messy .

 

[epenguin]

Fairly obviously you can just divide and get dY/dX independent of t, get a solution, use that to express dX/dt in terms of X alone.

Or you could differentiate (1) and RHS is dY/dt for which by eq. 2 you can express in terms of X and so get a 2nd order d.e.

Or you can see almost any textbook of math (or phys) that covers de's and get it more generally, if more longwindedly (I think it is easier to solve yrself and easier to read the books if you have done something your self).

(In fact this is nothing but the equations for simple harmonic motion - special case or special units isn't it, X displacement, Y momentum?)