60.2.127 problem 703

Internal problem ID [10714]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 703
Date solved : Monday, January 27, 2025 at 09:29:23 PM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }&=\frac {y \left (1-x +y x^{2} \ln \left (x \right )+x^{3} y-x \ln \left (x \right )-x^{2}\right )}{\left (x -1\right ) x} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 45

dsolve(diff(y(x),x) = y(x)*(1-x+y(x)*x^2*ln(x)+x^3*y(x)-x*ln(x)-x^2)/(x-1)/x,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{\operatorname {dilog}\left (x \right )-x}}{x \left (-\int \frac {{\mathrm e}^{\operatorname {dilog}\left (x \right )-x} \left (x +\ln \left (x \right )\right )}{\left (x -1\right )^{2}}d x +c_{1} \right ) \left (x -1\right )} \]

✓ Solution by Mathematica

Time used: 1.074 (sec). Leaf size: 168

DSolve[D[y[x],x] == (y[x]*(1 - x - x^2 - x*Log[x] + x^3*y[x] + x^2*Log[x]*y[x]))/((-1 + x)*x),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {e^{-\operatorname {PolyLog}(2,x)-x} (1-x)^{-\log (x)-1}}{x \left (-\int _1^xe^{-K[1]-\operatorname {PolyLog}(2,K[1])} (1-K[1])^{-\log (K[1])-2} (-K[1]-\log (K[1]))dK[1]+c_1\right )} \\ y(x)\to 0 \\ y(x)\to -\frac {e^{-\operatorname {PolyLog}(2,x)-x} (1-x)^{-\log (x)-1}}{x \int _1^xe^{-K[1]-\operatorname {PolyLog}(2,K[1])} (1-K[1])^{-\log (K[1])-2} (-K[1]-\log (K[1]))dK[1]} \\ \end{align*}