Internal problem ID [9375]
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.
CAS Maple gives this as type [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {\left (a^{2}-x^{2}\right ) \left (a^{2}-y^{2}\right ) y^{\prime \prime }+\left (a^{2}-x^{2}\right ) y \left (y^{\prime }\right )^{2}-x \left (a^{2}-y^{2}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 61
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 \relax (x ) = -a \\ y \relax (x ) = a \\ y \relax (x ) = \frac {\left (\left (x +\sqrt {-a^{2}+x^{2}}\right )^{2 c_{1}} c_{2}^{2}+a^{2}\right ) \left (x +\sqrt {-a^{2}+x^{2}}\right )^{-c_{1}}}{2 c_{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.178 (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*}