Internal problem ID [9276]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1697.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y^{\prime \prime } y+\left (y^{\prime }\right )^{2}-a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.092 (sec). Leaf size: 39
dsolve(diff(diff(y(x),x),x)*y(x)+diff(y(x),x)^2-a=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \sqrt {a \,x^{2}-2 x c_{1}+2 c_{2}} \\ y \relax (x ) = -\sqrt {a \,x^{2}-2 x c_{1}+2 c_{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.141 (sec). Leaf size: 68
DSolve[-a + y'[x]^2 + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {a^2 (x+c_2){}^2-e^{2 c_1}}}{\sqrt {a}} \\ y(x)\to \frac {\sqrt {a^2 (x+c_2){}^2-e^{2 c_1}}}{\sqrt {a}} \\ \end{align*}