Internal problem ID [15195]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 14. Differential equations admitting of
depression of their order. Exercises page 107
Problem number: 337.
ODE order: 1.
ODE degree: 0.
CAS Maple gives this as type [_separable]
\[ \boxed {y x -y^{\prime } \ln \left (\frac {y^{\prime }}{x}\right )=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 63
dsolve(x*y(x)=diff(y(x),x)*ln(diff(y(x),x)/x),y(x), singsol=all)
\begin{align*} y &= \left (-1-\sqrt {x^{2}-2 c_{1} +1}\right ) {\mathrm e}^{-1-\sqrt {x^{2}-2 c_{1} +1}} \\ y &= \left (-1+\sqrt {x^{2}-2 c_{1} +1}\right ) {\mathrm e}^{-1+\sqrt {x^{2}-2 c_{1} +1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.223 (sec). Leaf size: 83
DSolve[x*y[x]==y'[x]*Log[y'[x]/x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -e^{-1-\sqrt {x^2+1+2 c_1}} \left (1+\sqrt {x^2+1+2 c_1}\right ) \\ y(x)\to e^{-1+\sqrt {x^2+1+2 c_1}} \left (-1+\sqrt {x^2+1+2 c_1}\right ) \\ y(x)\to 0 \\ \end{align*}