60.2.340 problem 917

Internal problem ID [10927]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 917
Date solved : Tuesday, January 28, 2025 at 05:31:15 PM
CAS classification : [NONE]

\begin{align*} y^{\prime }&=\frac {y \left (x \ln \left (x \right )+\ln \left (x \right )+\ln \left (y\right ) x +\ln \left (y\right )-x -1+x \ln \left (x \right )^{2}+2 x \ln \left (y\right ) \ln \left (x \right )+x \ln \left (y\right )^{2}\right )}{x \left (1+x \right )} \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 38

dsolve(diff(y(x),x) = y(x)*(x*ln(x)+ln(x)+ln(y(x))*x+ln(y(x))-x-1+x*ln(x)^2+2*x*ln(y(x))*ln(x)+x*ln(y(x))^2)/x/(x+1),y(x), singsol=all)
 

\[ y = {\mathrm e}^{\frac {\ln \left (x \right ) \ln \left (x +1\right )+\left (-x +c_{1} \right ) \ln \left (x \right )-x}{-\ln \left (x +1\right )-c_{1} +x}} \]

✓ Solution by Mathematica

Time used: 0.492 (sec). Leaf size: 35

DSolve[D[y[x],x] == ((-1 - x + Log[x] + x*Log[x] + x*Log[x]^2 + Log[y[x]] + x*Log[y[x]] + 2*x*Log[x]*Log[y[x]] + x*Log[y[x]]^2)*y[x])/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {e^{-\frac {x}{x-\log (x+1)-c_1}}}{x} \\ y(x)\to \frac {1}{x} \\ \end{align*}