Change of variable in ODE

Question

In some papers about hydrodynamics in droplets, the following ODE

$g(h) + \frac{d^2 h}{dx^2} = p$

is commonly integrated to give

$f(h) + \frac{1}{2}\left(\frac{dh}{dx}\right)^2 = ph + c$

where $h=h(x)$, $g(h)=df/dh$, $p$ and $c$ are constants.

How is the second equation obtained from the first? Basically I'm quite sure how to integrate $h_{xx}$ to get $h_x^2/2$. How to do this properly/rigorously and what are the assumptions behind the steps?

Answer 1

Multiply the first equation with $\frac{dh}{dx}$ then integrate over $x$:

• First term $$\int dx\frac{dh}{dx}g(h)=\int dh g(h)=f(h)$$
• second term $$\int dx\frac{dh}{dx}\frac{d^2h}{dx^2}=\int\frac{dh}{dx}\left(\frac{dh}{dx}\right)'dx=\frac12\left(\frac{dh}{dx}\right)^2$$
• right hand side $$\int dx\frac{dh}{dx}p=\int p dh =ph+c$$