How Do You Solve a Second Order Nonlinear Autonomous ODE?

[MHD93]

Hel(lo, p)

I hope you're doing fine

I'm stuck with the following:

$y'' = -1/(y^2)$

I tried guessing functions (exponentials, roots, trigs... ) , but none worked, I haven't had any DE course, so I don't have specific steps to employ,

I appreciate your help,
Thanks in advance

 

[bigfooted]

the case where $y^{''}=f(y,y^{'})$ is called an autonomous equation (x does not occur directly in the right-hand side).

It can be solved by performing a simple transformation (this transformation follows from a translational symmetry of the ODE)

Let $y^{'}=z$, then
$y^{''}=z^{'}=\frac{dz}{dx}=\frac{dz}{dy}\frac{dy}{dx}=\frac{dz}{dy}z$
and the equation can be written as:
$z\frac{dz}{dy}=-\frac{1}{y^2}$
$\int z dz=-\int \frac{1}{y^2}dy$,
then transform back to the original variable y and integrate again.