Solving $f'''+\frac{n+1}{2}ff''-nf'^2+n=0$ with $n=e^\pi$

Question

How do I solve

$$f'''+\frac{n+1}{2}ff''-nf'^2+n=0$$

with $n=e^\pi$ or arbitrary $n$?

This equation occurs in my model for the time evolution of the value of Bitcoin.

Answer 1

This is a third order non-linear autonomous ODE. It can be reduced to a second order non-linear ODE. I think that it cannot be analytically solved in the general case with a finite number of elementary and/or current referenced special functions.! $$\text{Change of notation for clarity: $f(x)\equiv y(x)$}\\\,\\ \dfrac{\mathrm d^3y}{\mathrm dx^3}+\dfrac{n+1}{2}y\dfrac{\mathrm d^2y}{\mathrm dx}-n\left(\dfrac{\mathrm dy}{\mathrm dx}\right)^2+n=0\\\,\\ \text{Let : }\dfrac{\mathrm dy}{\mathrm dx}=F(y)\quad\to\quad\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm dF}{\mathrm dy}\dfrac{\mathrm dy}{\mathrm dx}=F\dfrac{\mathrm dF}{\mathrm dy}\\\,\\ \dfrac{\mathrm d^3y}{\mathrm dx^3}=\dfrac{\mathrm d}{\mathrm dy}\left(\dfrac{\mathrm d^2y}{\mathrm dx^2}\right)\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm d}{\mathrm dy}\left(F\dfrac{\mathrm dF}{\mathrm dy}\right)F=F^2\dfrac{\mathrm d^2F}{\mathrm dy^2}+F\left(\dfrac{\mathrm dF}{\mathrm dy}\right)^2\\\,\\ F^2\dfrac{\mathrm d^2F}{\mathrm dy^2}+F\left(\dfrac{\mathrm dF}{\mathrm dy}\right)^2+\dfrac{n+1}{2}yF\dfrac{\mathrm dF}{\mathrm dy}-nF^2+n=0 $$

Answer 2

Similar to How do you solve this kind of third order ODE? (non-linear Blasius equation),

Let $u=(f')^2$ ,

Then $\pm\dfrac{\sqrt u}{2}\dfrac{d^2u}{df^2}+\dfrac{n+1}{4}f\dfrac{du}{df}-nu+n=0$