Solution of $f(x)^2\frac{d^2}{dx^2}f(x)=x$

Question

I am stuck in finding the solution of this apparently simple differential equation: $$f(x)^2\dfrac{d^2}{dx^2}f(x)=x$$ with$f(0)=a$ and $f(0)'=b$ Using Maple the solution seems to be a combination of integrals of Airy functions, but not in a closed form. Is it possible to find something better than Maple solution? Thanks in advance.

Answer 1

Following this: http://eqworld.ipmnet.ru/en/solutions/ode/ode0310.pdf

Let $y = f(x), w = \frac{x}{y}y', z=\frac{x^3}{y^3}$, then $$w'_x = \frac{1}{y}y'+\frac{x}{y}y''-\frac{x}{y^2}(y')^2,$$ $$w'_x = w_z'\frac{dz}{dx}=w'_z (3\frac{x^2}{y^3}-3\frac{x^3}{y^4}y'_x),$$ $$xw_x'=3w_z'(z-zw),$$ $$xw_x'=\frac{x}{y}y'+\frac{x^2}{y}y''-\frac{x^2}{y^2}(y')^2=w+z-w^2,$$since $y''=\frac{x}{y^2}$. So, $3z(1-w)w_z'=z+w-w^2.$ Let $v=1-w$, $$3zvv_z'=z+(1-v)v,$$ $$vv_z'=\frac{1}{3}+\frac{v}{3z}-\frac{v^2}{3z}.$$ Then, we follow this: http://eqworld.ipmnet.ru/en/solutions/ode/ode0126.pdf

The result is messy.

Answer 2

This belongs to a special case of Emden-Fowler equation.

And luckily we can find its general solution in http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%20Ordinary%20Differential%20EquationsSecond%20Edition%5Cc2972_fm.pdf#page=333:

$\begin{cases}x=-\dfrac{\sqrt[3]9b}{\sqrt[3]2t^\frac{2}{3}}\left(\left(t\dfrac{d}{dt}\left(C_1J_\frac{1}{3}(t)+C_2Y_\frac{1}{3}(t)\right)+\dfrac{C_1}{3}J_\frac{1}{3}(t)+\dfrac{C_2}{3}Y_\frac{1}{3}(t)\right)^2+t^2\left(C_1J_\frac{1}{3}(t)+C_2Y_\frac{1}{3}(t)\right)^2\right)\\f=bt^\frac{2}{3}\left(C_1J_\frac{1}{3}(t)+C_2Y_\frac{1}{3}(t)\right)^2\end{cases}$

The reason of this special case of Emden-Fowler equation having the quite nice form of the general solution can be explained as according to http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%20Ordinary%20Differential%20EquationsSecond%20Edition%5Cc2972_fm.pdf#page=102, this ODE can be transformed as a second-order linear ODE.