75.14.11 problem 337

Internal problem ID [16846]
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 : Thursday, March 13, 2025 at 08:53:10 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.569 (sec). Leaf size: 63

ode:=x*y(x) = diff(y(x),x)*ln(diff(y(x),x)/x); 
dsolve(ode,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*}

✓ Mathematica. Time used: 0.166 (sec). Leaf size: 33

ode=x*y[x]==D[y[x],x]*Log[D[y[x],x]/x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.743 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x) - log(Derivative(y(x), x)/x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ - \frac {x^{2}}{2} + \frac {y{\left (x \right )} e^{- W\left (y{\left (x \right )}\right )} W\left (y{\left (x \right )}\right )}{2} + y{\left (x \right )} e^{- W\left (y{\left (x \right )}\right )} = C_{1} \]