Internal problem ID [9273]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1694.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {y^{\prime \prime } y-a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.096 (sec). Leaf size: 53
dsolve(diff(diff(y(x),x),x)*y(x)-a=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}\frac {1}{\sqrt {2 \ln \left (\textit {\_a} \right ) a -c_{1}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {1}{\sqrt {2 \ln \left (\textit {\_a} \right ) a -c_{1}}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.085 (sec). Leaf size: 111
DSolve[-a + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \exp \left (-\frac {2 a \text {erf}^{-1}\left (-i \sqrt {\frac {2}{\pi }} \sqrt {a e^{\frac {c_1}{a}} (x+c_2){}^2}\right ){}^2+c_1}{2 a}\right ) \\ y(x)\to \exp \left (-\frac {2 a \text {erf}^{-1}\left (i \sqrt {\frac {2}{\pi }} \sqrt {a e^{\frac {c_1}{a}} (x+c_2){}^2}\right ){}^2+c_1}{2 a}\right ) \\ \end{align*}