Converting a second-order ODE into system of first-order ODEs

[dreamspace]

This is not homework, but rather me just trying to work a numerical analysis problem.

I have a second order equation on the form m*y'' = a*y + n*x (no first derivative)

How does one convert this? It's been years since I did this. Last I remember, one would start with substituting the first derivative with something. (u = y'), but now there's no such part in the equation.

Thanks!

 

[haruspex]

Usually start by solving the homogeneous version.

 

[HallsofIvy]

I'm afraid haruspex misunderstood the question.

dreamspace, do exactly as you suggest: introduce the new dependent variable by defining z= y'. Then the equation becomes mz'= ay+ nx.

Your two first order equations are
y'= z
z'= (a/m)y+ (n/m)x