Internal problem ID [10118]
Internal file name [OUTPUT/9065_Monday_June_06_2022_06_20_19_AM_56593492/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1797 (book 6.206).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {\left (a^{2}-x^{2}\right ) \left (a^{2}-y^{2}\right ) y^{\prime \prime }+\left (a^{2}-x^{2}\right ) y {y^{\prime }}^{2}-x \left (a^{2}-y^{2}\right ) y^{\prime }=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville <- 2nd_order Liouville successful`
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 60
dsolve((a^2-x^2)*(a^2-y(x)^2)*diff(diff(y(x),x),x)+(a^2-x^2)*y(x)*diff(y(x),x)^2-x*(a^2-y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -a \\ y \left (x \right ) &= a \\ y \left (x \right ) &= \frac {\left (x +\sqrt {-a^{2}+x^{2}}\right )^{c_{1}} c_{2}^{2}+\left (x +\sqrt {-a^{2}+x^{2}}\right )^{-c_{1}} a^{2}}{2 c_{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.459 (sec). Leaf size: 195
DSolve[-(x*(a^2 - y[x]^2)*y'[x]) + (a^2 - x^2)*y[x]*y'[x]^2 + (a^2 - x^2)*(a^2 - y[x]^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{2} e^{-c_2} \left (\frac {a^2}{a^2-x^2}\right )^{-\frac {c_1}{2}} \sqrt {-a^2 \left (\left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{c_1}-e^{2 c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{c_1}\right ){}^2} \\ y(x)\to \frac {1}{2} e^{-c_2} \left (\frac {a^2}{a^2-x^2}\right )^{-\frac {c_1}{2}} \sqrt {-a^2 \left (\left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{c_1}-e^{2 c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{c_1}\right ){}^2} \\ \end{align*}