60.2.42 problem 618

Internal problem ID [10629]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 618
Date solved : Tuesday, January 28, 2025 at 05:00:18 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=\frac {\left (y+1\right ) \left (\left (y-\ln \left (y+1\right )-\ln \left (x \right )\right ) x +1\right )}{y x} \end{align*}

✓ Solution by Maple

Time used: 2.086 (sec). Leaf size: 36

dsolve(diff(y(x),x) = (y(x)+1)*((y(x)-ln(y(x)+1)-ln(x))*x+1)/y(x)/x,y(x), singsol=all)
 

\begin{align*} y &= -\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-1}}{x}\right )-1 \\ y &= -\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-1+{\mathrm e}^{x} c_{1}}}{x}\right )-1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.184 (sec). Leaf size: 25

DSolve[D[y[x],x] == ((1 + y[x])*(1 + x*(-Log[x] - Log[1 + y[x]] + y[x])))/(x*y[x]),y[x],x,IncludeSingularSolutions -> True]
 

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