Internal problem ID [10852]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {x y y^{\prime \prime }-x {y^{\prime }}^{2}-y^{\prime } y=0} \]
✓ Solution by Maple
Time used: 0.015 (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 \left (x \right ) = 0 \\ y \left (x \right ) = {\mathrm e}^{\frac {c_{1} x^{2}}{2}} c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.031 (sec). Leaf size: 19
DSolve[x*y[x]*y''[x]-x*y'[x]^2-y[x]*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 e^{\frac {c_1 x^2}{2}} \\ \end{align*}