66.2.22 problem Problem 31

Internal problem ID [13921]
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
Date solved : Tuesday, January 28, 2025 at 06:08:38 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y y^{\prime } y^{\prime \prime }&={y^{\prime }}^{3}+{y^{\prime \prime }}^{2} \end{align*}

✓ Solution by Maple

Time used: 0.237 (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 &= -\frac {4}{-4 c_{1} +x} \\ y &= c_{1} \\ y &= {\mathrm e}^{-\left (x +c_{2} \right ) c_{1}}-c_{1} \\ y &= {\mathrm e}^{\left (x +c_{2} \right ) c_{1}}+c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 8.330 (sec). Leaf size: 119

DSolve[y[x]*D[y[x],x]*D[y[x],{x,2}]==D[y[x],x]^3+D[y[x],{x,2}]^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*}