How do you convert second order differential equations to first order differential equations?

Answer 1

You can't convert a second order DE to first order except in special cases (like an ODE with y'' and y' but no y terms).

HOWEVER, you can convert a second order ODE into a systemof first order ODEs:

Assume it is of the form f(x, y, y', y'') = 0, where y(x) is the solution.

1) Let u1 = y and u2 = y'

2) Substitute y'' for u2', y' for u2, and y for u1 to get eq1

3) u2 = u1' is eq2.

eq1 and eq2 are a system of two first-order ODEs which represent the same problem.