Internal problem ID [9343]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1765 (book 6.174).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {x y y^{\prime \prime }-2 x \left (y^{\prime }\right )^{2}+\left (y+1\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 22
dsolve(x*y(x)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)^2+(y(x)+1)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = c_{1} \tanh \left (\frac {\ln \relax (x )-c_{2}}{2 c_{1}}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.046 (sec). Leaf size: 37
DSolve[(1 + y[x])*y'[x] - 2*x*y'[x]^2 + x*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\tan \left (\frac {\sqrt {c_1} (\log (x)-c_2)}{\sqrt {2}}\right )}{\sqrt {2} \sqrt {c_1}} \\ \end{align*}