75.14.11 problem 337

Internal problem ID [16925]
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 ODEs). Section 14. Differential equations admitting of depression of their order. Exercises page 107
Problem number : 337
Date solved : Tuesday, January 28, 2025 at 09:41:03 AM
CAS classification : [_separable]

\begin{align*} y x&=y^{\prime } \ln \left (\frac {y^{\prime }}{x}\right ) \end{align*}

✓ Solution by Maple

Time used: 0.569 (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: 0.166 (sec). Leaf size: 33

DSolve[x*y[x]==D[y[x],x]*Log[D[y[x],x]/x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {W(K[1])}{K[1]}dK[1]\&\right ]\left [\frac {x^2}{2}+c_1\right ] \]