How Do You Solve Autonomous Second Order ODEs Like y'' = f(y)?

[muzialis]

Hi All,

I was looking for the general solution of an equation as y(x)'' = f (y), and found the attached document on the web.

It presents the solution in a way which I am not sure I understand. I tried to look at the trivila example y'' = - y, solution y = sin (x), but I am struggling in obtaining this result with the provided solution.

If anybody could help, I would be the most obliged.

All the Best

Muzialis

 

[mathador]

Hi,

The trick is to define a new variable, y'=dy/dx=p(y(x)).
Now y''=dp/dy*y'=dp/dy*p=f(y).
Integrating now yields p^2=c+int(f(y))dy. The next step is to integrate dx=dy/p(y).

Mathador