66.2.44 problem Problem 59

Internal problem ID [13943]
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 59
Date solved : Tuesday, January 28, 2025 at 06:09:19 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.115 (sec). Leaf size: 20

dsolve(y(x)*diff(y(x),x$2)-diff(y(x),x)^2=diff(y(x),x),y(x), singsol=all)
 

\begin{align*} y &= 0 \\ y &= \frac {{\mathrm e}^{\left (x +c_{2} \right ) c_{1}}+1}{c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.469 (sec). Leaf size: 26

DSolve[y[x]*D[y[x],{x,2}]-D[y[x],x]^2==D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1+e^{c_1 (x+c_2)}}{c_1} \\ y(x)\to \text {Indeterminate} \\ \end{align*}