Internal problem ID [10134]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1812 (book 6.221).
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\sqrt {y^{2}+x^{2}}\, y^{\prime \prime }-a \left ({y^{\prime }}^{2}+1\right )^{\frac {3}{2}}=0} \]
✓ Solution by Maple
Time used: 0.188 (sec). Leaf size: 88
dsolve((y(x)^2+x^2)^(1/2)*diff(diff(y(x),x),x)-a*(diff(y(x),x)^2+1)^(3/2)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {-\operatorname {RootOf}\left (\arctan \left (\textit {\_g} \right )+\int _{}^{\textit {\_Z}}\frac {1+\sqrt {a^{2} \left (\textit {\_f}^{2}+1\right )}}{\left (\textit {\_f}^{2} a^{2}+a^{2}-1\right ) \left (\textit {\_f}^{2}+1\right )}d \textit {\_f} +c_{1} \right )+\textit {\_g}}{\textit {\_g}^{2}+1}d \textit {\_g} +c_{2} \right ) x \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[-(a*(1 + y'[x]^2)^(3/2)) + Sqrt[x^2 + y[x]^2]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
Timed out