ordinary differential equations
$ y_n'' -nx\frac{1}{\sqrt {y_n}}=0$
Is there any known method to solve such second order non-linear differential equation?
What I tried to solve:
$ 2y_n'y_n''= 2nx\frac{y_n'}{\sqrt {y_n}}$
$ y_n'^2= 4nx\sqrt {y_n}-4n\int\sqrt {y_n} dx$
After that I could not see any way how to proceed.
Please advice what to do to solve the differential equation.
Many thanks
One can express $t$ as a function of $x$, function defined by an integral (no simpler closed form).
There is no closed form for the inverse function $x(t)$.
On practical viewpoint, better use a numerical method for solving.
A particular solution is obviously a degree $1$ polynomial: $y_0=at+b$. Since $y_0'=a$ and $y_0''=0$, the relation yields $4a+at+b=t$, that is, $a=1$ and $b=-4$.
For the general solution the general method does work! When the characteristic polynomial has a root $\lambda$ of multiplicity $m$, you get linearly independent solutions of the form $t^{k}e^{\lambda t}$, for $k=0,1,\dots,m-1$.
In this case, you get $y_1=e^{-2t}$ and $y_2=te^{-2t}$.
Best Answer
In fact it belongs to an Emden-Fowler equation.
First, according to http://eqworld.ipmnet.ru/en/solutions/ode/ode0302.pdf or http://www.ae.illinois.edu/lndvl/Publications/2002_IJND.pdf#page=6 , all Emden-Fowler equations can be transformed into Abel equation of the second kind.
Let $\begin{cases}u=\dfrac{x^3}{y_n^\frac{3}{2}}\\v=\dfrac{x}{y_n}\dfrac{dy_n}{dx}\end{cases}$ ,
Then $\dfrac{dv}{du}=\dfrac{\dfrac{dv}{dx}}{\dfrac{du}{dx}}=\dfrac{\dfrac{x}{y_n}\dfrac{d^2y_n}{dx^2}+\dfrac{1}{y_n}\dfrac{dy_n}{dx}-\dfrac{x}{y_n^2}\left(\dfrac{dy_n}{dx}\right)^2}{\dfrac{3x^2}{y_n^\frac{3}{2}}-\dfrac{3x^3}{2y_n^\frac{5}{2}}\dfrac{dy_n}{dx}}=\dfrac{\dfrac{x}{y_n}\dfrac{d^2y_n}{dx^2}+\dfrac{v}{x}-\dfrac{v^2}{x}}{\dfrac{3u}{x}-\dfrac{3uv}{2x}}=\dfrac{\dfrac{x^2}{y_n}\dfrac{d^2y_n}{dx^2}+v-v^2}{3u\left(1-\dfrac{v}{2}\right)}$
$3u\left(1-\dfrac{v}{2}\right)\dfrac{dv}{du}=\dfrac{x^2}{y_n}\dfrac{d^2y_n}{dx^2}+v-v^2$
$\dfrac{x^2}{y_n}\dfrac{d^2y_n}{dx^2}=3u\left(1-\dfrac{v}{2}\right)\dfrac{dv}{du}+v^2-v$
$\dfrac{d^2y_n}{dx^2}=\dfrac{y_n}{x^2}\left(3u\left(1-\dfrac{v}{2}\right)\dfrac{dv}{du}+v^2-v\right)$
$\therefore\dfrac{y_n}{x^2}\left(3u\left(1-\dfrac{v}{2}\right)\dfrac{dv}{du}+v^2-v\right)-\dfrac{nx}{\sqrt{y_n}}=0$
$\dfrac{y_n}{x^2}\left(3u\left(1-\dfrac{v}{2}\right)\dfrac{dv}{du}+v^2-v\right)=\dfrac{nx}{\sqrt{y_n}}$
$3u\left(1-\dfrac{v}{2}\right)\dfrac{dv}{du}+v^2-v=\dfrac{nx^3}{y_n^\frac{3}{2}}$
$3u\left(1-\dfrac{v}{2}\right)\dfrac{dv}{du}+v^2-v=nu$
$3u\left(\dfrac{v}{2}-1\right)\dfrac{dv}{du}=v^2-v-nu$
Let $w=\dfrac{v}{2}-1$ ,
Then $v=2w+2$
$\dfrac{dv}{du}=2\dfrac{dw}{du}$
$\therefore6uw\dfrac{dw}{du}=(2w+2)^2-(2w+2)-nu$
$6uw\dfrac{dw}{du}=4w^2+6w+2-nu$
$w\dfrac{dw}{du}=\dfrac{2w^2}{3u}+\dfrac{w}{u}+\dfrac{2-nu}{6u}$
In fact, all Abel equation of the second kind can be transformed into Abel equation of the first kind.
Let $w=\dfrac{1}{z}$ ,
Then $\dfrac{dw}{du}=-\dfrac{1}{z^2}\dfrac{dz}{du}$
$\therefore-\dfrac{1}{z^3}\dfrac{dz}{du}=\dfrac{2}{3uz^2}+\dfrac{1}{uz}+\dfrac{2-nu}{6u}$
$\dfrac{dz}{du}=\dfrac{(nu-2)z^3}{6u}-\dfrac{z^2}{u}-\dfrac{2z}{3u}$
Please follow the method in http://www.hindawi.com/journals/ijmms/2011/387429/#sec2.