66.2.41 problem Problem 56

Internal problem ID [13940]
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 : Tuesday, January 28, 2025 at 06:09:13 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

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

dsolve(x*diff(y(x),x$2)=diff(y(x),x)*ln(diff(y(x),x)/x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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