Internal problem ID [9403]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1825 (book 6.234).
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_y]]
Solve \begin {gather*} \boxed {\left (a \sqrt {\left (y^{\prime }\right )^{2}+1}-y^{\prime } x \right ) y^{\prime \prime }-\left (y^{\prime }\right )^{2}-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.422 (sec). Leaf size: 117
dsolve((a*(diff(y(x),x)^2+1)^(1/2)-x*diff(y(x),x))*diff(diff(y(x),x),x)-diff(y(x),x)^2-1=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -i x +c_{1} \\ y \relax (x ) = i x +c_{1} \\ y \relax (x ) = \int \frac {-c_{1} a^{2}+x \sqrt {a^{2} \left (a^{2}-x^{2}+c_{1}^{2}\right )}}{a \left (a^{2}-x^{2}\right )}d x +c_{2} \\ y \relax (x ) = \int -\frac {c_{1} a^{2}+x \sqrt {a^{2} \left (a^{2}-x^{2}+c_{1}^{2}\right )}}{a \left (a^{2}-x^{2}\right )}d x +c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.507 (sec). Leaf size: 315
DSolve[-1 - y'[x]^2 + (-(x*y'[x]) + a*Sqrt[1 + y'[x]^2])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 x-\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}}{x}+\frac {1}{2} c_1 \left (\log \left (-c_1 \left (\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x\right )+a^2 (-x)+a x^2\right )+\log \left (c_1 \left (\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x\right )+a^2 x+a x^2\right )+\log (x-a)-\log (x (x-a))-\log (a+x)-\log (x (a+x))\right ) \\ y(x)\to \frac {\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_2 x}{x}+\frac {1}{2} c_1 \left (-\log \left (-c_1 \left (\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x\right )+a^2 (-x)+a x^2\right )-\log \left (c_1 \left (\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x\right )+a^2 x+a x^2\right )+\log (x-a)+\log (x (x-a))-\log (a+x)+\log (x (a+x))\right ) \\ \end{align*}