66.2.37 problem Problem 52

Internal problem ID [13936]
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 52
Date solved : Tuesday, January 28, 2025 at 06:09:11 AM
CAS classification : [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x y y^{\prime \prime }-x {y^{\prime }}^{2}-y^{\prime } y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 17

dsolve(x*y(x)*diff(y(x),x$2)-x*diff(y(x),x)^2-y(x)*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ y &= {\mathrm e}^{\frac {c_{1} x^{2}}{2}} c_{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.118 (sec). Leaf size: 19

DSolve[x*y[x]*D[y[x],{x,2}]-x*D[y[x],x]^2-y[x]*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 e^{\frac {c_1 x^2}{2}} \]