Internal problem ID [15218]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 14. Differential equations admitting of
depression of their order. Exercises page 107
Problem number: 360.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type
[[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {y y^{\prime \prime }-{y^{\prime }}^{2}-y^{\prime } y^{2}=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 27
dsolve(y(x)*diff(y(x),x$2)-diff(y(x),x)^2=y(x)^2*diff(y(x),x),y(x), singsol=all)
\begin{align*} y &= 0 \\ y &= -\frac {c_{1} {\mathrm e}^{c_{1} \left (x +c_{2} \right )}}{-1+{\mathrm e}^{c_{1} \left (x +c_{2} \right )}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.39 (sec). Leaf size: 43
DSolve[y[x]*y''[x]-y'[x]^2==y[x]^2*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {c_1 e^{c_1 (x+c_2)}}{-1+e^{c_1 (x+c_2)}} \\ y(x)\to -\frac {1}{x+c_2} \\ \end{align*}