Internal problem ID [9346]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1768 (book 6.177).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y y^{\prime \prime }+\left (\frac {a x}{\sqrt {b^{2}-x^{2}}}-x \right ) \left (y^{\prime }\right )^{2}-y^{\prime } y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.25 (sec). Leaf size: 54
dsolve(x*y(x)*diff(diff(y(x),x),x)+(a*x/(b^2-x^2)^(1/2)-x)*diff(y(x),x)^2-y(x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = c_{2} {\mathrm e}^{\int -\frac {x \sqrt {b^{2}-x^{2}}}{a \,b^{2}-a \,x^{2}+c_{1} \sqrt {b^{2}-x^{2}}}d x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.217 (sec). Leaf size: 54
DSolve[-(y[x]*y'[x]) + (-x + (a*x)/Sqrt[b^2 - x^2])*y'[x]^2 + x*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 e^{\frac {\sqrt {b^2-x^2}}{a}} \left (a \sqrt {(b-x) (b+x)}-c_1\right ){}^{\frac {c_1}{a^2}} \\ \end{align*}