Is there any known method to solve such second order

Minerva Kline 2021-11-20
Is there any known method to solve such second order non-linear differential equation?
\(\displaystyle{y}{''}_{{n}}-{n}{x}{\frac{{{1}}}{{\sqrt{{{y}_{{n}}}}}}}={0}\)

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Answered 2021-11-25 Author has 1938 answers

Let \(\begin{cases}u=\frac{x^3}{y_n^{\frac32}}\\v=\frac{x}{y_n}\frac{dy_n}{dx}\end{cases}\)
Then
\(\frac{dv}{du}=\frac{\frac{dv}{dx}}{\frac{du}{dx}}\)
\(=\frac{\frac{x}{y_n}\frac{d^2y_n}{dx^2}+\frac{1}{y_n}\frac{dy_n}{dx}-\frac{x}{y_n^2}(\frac{dy_n}{dx})^2}{\frac{3x^2}{y_n^{\frac{3}{2}}}-\frac{3x^3}{2y_n^{\frac{5}{2}}}\frac{dy_n}{dx}}\)
\(=\frac{\frac{x}{y_n}\frac{d^2y_n}{dx^2}+\frac{v}{x}-\frac{v^2}{x}}{\frac{3u}{x}-\frac{3uv}{2x}}\)
\(=\frac{\frac{x^2}{y_n}\frac{d^2y_n}{dx^2}+v-v^2}{3u(1-\frac{v}{2})}\)
\(3u(1-\frac{v}{2})\frac{dv}{du}=\frac{x^2}{y_n}\frac{d^2y_n}{dx^2}+v-v^2\)
\(\frac{x^2}{y_n}\frac{d^2y_n}{dx^2}=3u(1-\frac{v}{2})\frac{du}{dv}+v^2-v\)
\(\frac{d^2y_n}{dx^2}=\frac{y_n}{x^2}(3u(1-\frac{v}{2})\frac{dv}{du}+v^2-v)\)
\(\frac{y_n}{x^2}(3u(1-\frac{v}{2})\frac{dv}{du}+v^2-v)-\frac{nx}{\sqrt{y_n}}=0\)
\(\frac{y_n}{x^2}(3u(1-\frac{v}{2})\frac{dv}{du}+v^2-v)=\frac{nx}{\sqrt{y_n}}\)
\(3u(1-\frac{v}{2})\frac{dv}{du}+v^2-v=\frac{nx^3}{y_n^{\frac32}}\)
\(3u(1-\frac{v}{2})\frac{dv}{du}+v^2-v=nu\)
\(3u(\frac{v}{2}-1)\frac{dv}{du}=v^2-v-nu\)
Let \(w=\frac{w}{2}-1\)
Let \(w=\frac{v}{2}-1\)
Then \(v=2w+2\)
\(\frac{dv}{du}=2\frac{dw}{du}\)
\(6uw\frac{dw}{du}=4w^2+6w+2-nu\)
\(w\frac{dw}{du}=\frac{2w^2}{3u}+\frac{w}{u}+\frac{2-nu}{6u}\)
In fact, all Abel equation of the second kind can be transformed into Abel equation of the first kind.
NSK
Let \(w=\frac{1}{z}\)
Then \(\frac{dw}{du}=-\frac{1}{z^2}\frac{dz}{du}\)
\(-\frac{1}{z^3}\frac{dz}{du}=\frac{2}{3uz^2}+\frac{1}{uz}+\frac{2-nu}{6u}\)
\(\frac{dz}{du}=\frac{2}{3uz^2}+\frac{1}{uz}+\frac{2-nu}{6u}\)
\(\frac{dz}{du}=\frac{(nu-2)z^3}{6u}-\frac{z^2}{u}-\frac{2z}{3u}\)

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