60.2.125 problem 701

Internal problem ID [10712]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 701
Date solved : Tuesday, January 28, 2025 at 05:06:31 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=\frac {2 x \,{\mathrm e}^{x}-2 x -\ln \left (x \right )-1+x^{4} \ln \left (x \right )+x^{4}-2 y x^{2} \ln \left (x \right )-2 x^{2} y+y^{2} \ln \left (x \right )+y^{2}}{{\mathrm e}^{x}-1} \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 71

dsolve(diff(y(x),x) = (2*x*exp(x)-2*x-ln(x)-1+x^4*ln(x)+x^4-2*y(x)*x^2*ln(x)-2*x^2*y(x)+y(x)^2*ln(x)+y(x)^2)/(exp(x)-1),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.487 (sec). Leaf size: 97

DSolve[D[y[x],x] == (-1 - 2*x + 2*E^x*x + x^4 - Log[x] + x^4*Log[x] - 2*x^2*y[x] - 2*x^2*Log[x]*y[x] + y[x]^2 + Log[x]*y[x]^2)/(-1 + E^x),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\exp \left (\int _1^x\frac {2 (\log (K[5])+1)}{-1+e^{K[5]}}dK[5]\right )}{-\int _1^x\frac {\exp \left (\int _1^{K[6]}\frac {2 (\log (K[5])+1)}{-1+e^{K[5]}}dK[5]\right ) (\log (K[6])+1)}{-1+e^{K[6]}}dK[6]+c_1}+x^2+1 \\ y(x)\to x^2+1 \\ \end{align*}