Internal problem ID [9412]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1833.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {\left (a^{2} y^{2}-b^{2}\right ) \left (y^{\prime \prime }\right )^{2}-2 a^{2} y \left (y^{\prime }\right )^{2} y^{\prime \prime }+\left (a^{2} \left (y^{\prime }\right )^{2}-1\right ) \left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.111 (sec). Leaf size: 162
dsolve((a^2*y(x)^2-b^2)*diff(diff(y(x),x),x)^2-2*a^2*y(x)*diff(y(x),x)^2*diff(diff(y(x),x),x)+(a^2*diff(y(x),x)^2-1)*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\tan \left (\frac {\sqrt {a^{2}}\, \left (c_{1}-x \right )}{a b}\right ) b}{\sqrt {\tan ^{2}\left (\frac {\sqrt {a^{2}}\, \left (c_{1}-x \right )}{a b}\right )+1}\, a} \\ y \relax (x ) = -\frac {\tan \left (\frac {\sqrt {a^{2}}\, \left (c_{1}-x \right )}{a b}\right ) b}{\sqrt {\tan ^{2}\left (\frac {\sqrt {a^{2}}\, \left (c_{1}-x \right )}{a b}\right )+1}\, a} \\ y \relax (x ) = -\frac {b}{a} \\ y \relax (x ) = \frac {b}{a} \\ y \relax (x ) = c_{1} \\ y \relax (x ) = \frac {b \left ({\mathrm e}^{\frac {\sqrt {a^{2} c_{1}^{2}-1}\, \left (c_{2}+x \right )}{b}}-c_{1}\right )}{\sqrt {a^{2} c_{1}^{2}-1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 3.203 (sec). Leaf size: 81
DSolve[y'[x]^2*(-1 + a^2*y'[x]^2) - 2*a^2*y[x]*y'[x]^2*y''[x] + (-b^2 + a^2*y[x]^2)*y''[x]^2 == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {b \left (e^{\frac {\sqrt {-1+a^2 c_1{}^2} (x+c_2)}{b}}-c_1\right )}{\sqrt {-1+a^2 c_1{}^2}} \\ y(x)\to c_1 e^{c_2 x}-\frac {\sqrt {b^2+\frac {1}{c_2{}^2}}}{a} \\ \end{align*}