60.1.562 problem 565

Internal problem ID [10576]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 565
Date solved : Monday, January 27, 2025 at 09:16:23 PM
CAS classification : [_separable]

\begin{align*} y \ln \left (y^{\prime }\right )+y^{\prime }-y \ln \left (y\right )-y x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.217 (sec). Leaf size: 17

dsolve(y(x)*ln(diff(y(x),x))+diff(y(x),x)-y(x)*ln(y(x))-x*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} {\mathrm e}^{\frac {\operatorname {LambertW}\left ({\mathrm e}^{x}\right ) \left (\operatorname {LambertW}\left ({\mathrm e}^{x}\right )+2\right )}{2}} \]

✓ Solution by Mathematica

Time used: 0.090 (sec). Leaf size: 22

DSolve[-(x*y[x]) - Log[y[x]]*y[x] + Log[D[y[x],x]]*y[x] + D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \exp \left (\int _1^xW\left (e^{K[1]}\right )dK[1]\right ) \]