Internal problem ID [12185]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER.
Problems page 172
Problem number: Problem 31.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {y y^{\prime } y^{\prime \prime }-{y^{\prime }}^{3}-{y^{\prime \prime }}^{2}=0} \]
✓ Solution by Maple
Time used: 7.281 (sec). Leaf size: 42
dsolve(y(x)*diff(y(x),x)*diff(y(x),x$2)=diff(y(x),x)^3+diff(y(x),x$2)^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {4}{-4 c_{1} +x} \\ y \left (x \right ) &= c_{1} \\ y \left (x \right ) &= {\mathrm e}^{-c_{1} \left (c_{2} +x \right )}-c_{1} \\ y \left (x \right ) &= {\mathrm e}^{c_{1} \left (c_{2} +x \right )}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 13.794 (sec). Leaf size: 119
DSolve[y[x]*y'[x]*y''[x]==y'[x]^3+y''[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (e^{-\frac {1}{2} \left (1+e^{c_1}\right ) (x+c_2)}-1-e^{c_1}\right ) \\ y(x)\to \frac {1+e^{\frac {x+c_2}{-1+\tanh \left (\frac {c_1}{2}\right )}}}{-1+\tanh \left (\frac {c_1}{2}\right )} \\ y(x)\to -\frac {1}{2}-\frac {1}{2} e^{-\frac {x}{2}-\frac {c_2}{2}} \\ y(x)\to \frac {1}{2} \left (-1+e^{-\frac {x}{2}-\frac {c_2}{2}}\right ) \\ \end{align*}