Internal problem ID [9231]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1652.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-a \sqrt {y^{2} b +\left (y^{\prime }\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.162 (sec). Leaf size: 40
dsolve(diff(diff(y(x),x),x)-a*(y(x)^2*b+diff(y(x),x)^2)^(1/2)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = {\mathrm e}^{\int \RootOf \left (x -\left (\int _{}^{\textit {\_Z}}\frac {1}{-\textit {\_f}^{2}+a \sqrt {\textit {\_f}^{2}+b}}d \textit {\_f} \right )+c_{1}\right )d x +c_{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.349 (sec). Leaf size: 76
DSolve[-(a*Sqrt[b*y[x]^2 + y'[x]^2]) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {InverseFunction}\left [\int \frac {\text {$\#$1}}{K[1] \left (\frac {\text {$\#$1}^2}{K[1]^2}-a \sqrt {\frac {\text {$\#$1}^2}{K[1]^2}+b}\right )}d\frac {\text {$\#$1}}{K[1]}\&\right ][c_1-\log (K[1])]}dK[1]=x-c_2,y(x)\right ] \]