Internal problem ID [9280]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1702 (book 6.111).
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-\left (y^{\prime }\right )^{2}-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 79
dsolve(diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2-1=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left ({\mathrm e}^{-\frac {2 c_{2}}{c_{1}}} {\mathrm e}^{-\frac {2 x}{c_{1}}}+1\right ) {\mathrm e}^{\frac {c_{2}}{c_{1}}} {\mathrm e}^{\frac {x}{c_{1}}} c_{1}}{2} \\ y \relax (x ) = \frac {\left ({\mathrm e}^{\frac {2 c_{2}}{c_{1}}} {\mathrm e}^{\frac {2 x}{c_{1}}}+1\right ) {\mathrm e}^{-\frac {c_{2}}{c_{1}}} {\mathrm e}^{-\frac {x}{c_{1}}} c_{1}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.14 (sec). Leaf size: 80
DSolve[-1 - y'[x]^2 + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {e^{-c_1} \tanh \left (e^{c_1} (x+c_2)\right )}{\sqrt {-\text {sech}^2\left (e^{c_1} (x+c_2)\right )}} \\ y(x)\to \frac {e^{-c_1} \tanh \left (e^{c_1} (x+c_2)\right )}{\sqrt {-\text {sech}^2\left (e^{c_1} (x+c_2)\right )}} \\ \end{align*}