# second order non linear nonhomogeneous differential equation

I'm learning about 2nd order differential equation atm, and I was wondering if there is any kind of a general way to solve a nonlinear, nonhomogeneous second order differential equation.

For example, I'm working on this equation: $$y''+ A(y')^2 = B.$$ $$A$$ and $$B$$ are constants here.

I have absolutely no clue on how to approach this, so could someone please help me? Thanks.

## 2 Answers

Substitute $$v=y'$$ to obtain $$v'+Av^2=B.$$ You can separate: \begin{align*} v'&=B-Av^2\\ \frac{dv}{B-Av^2}&=dx. \end{align*} Integrate once, substitute in for $$y$$ and integrate again.

There's no general method, I'm afraid.

Answer: $$y(x)=\frac{\ln \left(\cosh \left(\sqrt{A} \sqrt{B} (c_1+x)\right)\right)}{A}+c_2.$$

There's no general nonlinear theory. Not even for equations of first degree. You have to deal with them as they come.