Internal problem ID [9282]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1703.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y^{\prime \prime } y-\left (y^{\prime }\right )^{2}-y^{2} \ln \relax (y)=0} \end {gather*}
✓ Solution by Maple
Time used: 0.014 (sec). Leaf size: 21
dsolve(diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2-y(x)^2*ln(y(x))=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{\frac {{\mathrm e}^{-2 x} c_{1} {\mathrm e}^{x}}{2}} {\mathrm e}^{-\frac {c_{2} {\mathrm e}^{x}}{2}} \]
✓ Solution by Mathematica
Time used: 0.264 (sec). Leaf size: 65
DSolve[-(Log[y[x]]*y[x]^2) - y'[x]^2 + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{\sinh \left (\sqrt {c_1} \sinh (x+c_2)\right )+\cosh \left (\sqrt {c_1} \sinh (x+c_2)\right )} \\ y(x)\to \sinh \left (\sqrt {c_1} \sinh (x+c_2)\right )+\cosh \left (\sqrt {c_1} \sinh (x+c_2)\right ) \\ \end{align*}