66.2.41 problem Problem 56

Internal problem ID [13861]
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 56
Date solved : Wednesday, March 05, 2025 at 10:19:53 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 31

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

\[ y = \frac {{\mathrm e}^{c_{1} x +1} c_{1} x +c_{2} c_{1}^{2}-{\mathrm e}^{c_{1} x +1}}{c_{1}^{2}} \]

✓ Mathematica. Time used: 0.534 (sec). Leaf size: 31

ode=x*D[y[x],{x,2}]==D[y[x],x]*Log[D[y[x],x]/x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{e^{c_1} x+1-2 c_1} \left (-1+e^{c_1} x\right )+c_2 \]

✓ Sympy. Time used: 1.057 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) - log(Derivative(y(x), x)/x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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