Internal problem ID [8281]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 701.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]], _Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {2 x \,{\mathrm e}^{x}-2 x -\ln \relax (x )-1+x^{4} \ln \relax (x )+x^{4}-2 y x^{2} \ln \relax (x )-2 x^{2} y+y^{2} \ln \relax (x )+y^{2}}{{\mathrm e}^{x}-1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 100
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)
\[ y \relax (x ) = \frac {x^{2} c_{1} {\mathrm e}^{\int -\frac {2}{\frac {{\mathrm e}^{x}}{\ln \relax (x )+1}-\frac {1}{\ln \relax (x )+1}}d x}-x^{2}+c_{1} {\mathrm e}^{\int -\frac {2}{\frac {{\mathrm e}^{x}}{\ln \relax (x )+1}-\frac {1}{\ln \relax (x )+1}}d x}+1}{c_{1} {\mathrm e}^{\int -\frac {2}{\frac {{\mathrm e}^{x}}{\ln \relax (x )+1}-\frac {1}{\ln \relax (x )+1}}d x}-1} \]
✓ Solution by Mathematica
Time used: 2.764 (sec). Leaf size: 97
DSolve[y'[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*}