Internal problem ID [9214]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1635.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+a \left (y^{\prime }\right )^{2}+b y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.029 (sec). Leaf size: 79
dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)^2+b*y(x)=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}-\frac {2 a}{\sqrt {4 \,{\mathrm e}^{-2 \textit {\_a} a} c_{1} a^{2}-4 \textit {\_a} a b +2 b}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {2 a}{\sqrt {4 \,{\mathrm e}^{-2 \textit {\_a} a} c_{1} a^{2}-4 \textit {\_a} a b +2 b}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.398 (sec). Leaf size: 104
DSolve[b*y[x] + a*y'[x]^2 + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {2} a}{\sqrt {2 e^{-2 a K[1]} c_1 a^2-2 b K[1] a+b}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {2} a}{\sqrt {2 e^{-2 a K[2]} c_1 a^2-2 b K[2] a+b}}dK[2]\&\right ][x+c_2] \\ \end{align*}