12.5.32 problem 29

Internal problem ID [1656]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 29
Date solved : Monday, January 27, 2025 at 05:18:18 AM
CAS classification : [[_homogeneous, `class A`]]

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

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 24

dsolve((diff(y(x),x)*x-y(x))*(ln(y(x))-ln(x))=x,y(x), singsol=all)
 

\[ y = \frac {x \ln \left (\frac {x}{c_1}\right )}{\operatorname {LambertW}\left (\ln \left (\frac {x}{c_1}\right ) {\mathrm e}^{-1}\right )} \]

✓ Solution by Mathematica

Time used: 60.161 (sec). Leaf size: 24

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

\[ y(x)\to \frac {x (\log (x)+c_1)}{W\left (\frac {\log (x)+c_1}{e}\right )} \]