Internal problem ID [9049]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 714.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
\[ \boxed {y^{\prime }+\frac {y \left (-\ln \left (\frac {1}{x}\right )+{\mathrm e}^{x}+y x^{2} \ln \left (x \right )+y x^{3}-x \ln \left (x \right )-x^{2}\right )}{\left (-\ln \left (\frac {1}{x}\right )+{\mathrm e}^{x}\right ) x}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 96
dsolve(diff(y(x),x) = -y(x)*(-ln(1/x)+exp(x)+y(x)*x^2*ln(x)+x^3*y(x)-x*ln(x)-x^2)/(-ln(1/x)+exp(x))/x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{\int \frac {x \ln \left (x \right )+x^{2}-{\mathrm e}^{x}+\ln \left (\frac {1}{x}\right )}{x \left (-\ln \left (\frac {1}{x}\right )+{\mathrm e}^{x}\right )}d x}}{\int \frac {{\mathrm e}^{\int \frac {x \ln \left (x \right )+x^{2}-{\mathrm e}^{x}+\ln \left (\frac {1}{x}\right )}{x \left (-\ln \left (\frac {1}{x}\right )+{\mathrm e}^{x}\right )}d x} x \left (\ln \left (x \right )+x \right )}{-\ln \left (\frac {1}{x}\right )+{\mathrm e}^{x}}d x +c_{1}} \]
✓ Solution by Mathematica
Time used: 2.236 (sec). Leaf size: 290
DSolve[y'[x] == -((y[x]*(E^x - x^2 - Log[x^(-1)] - x*Log[x] + x^3*y[x] + x^2*Log[x]*y[x]))/(x*(E^x - Log[x^(-1)]))),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\exp \left (\int _1^x\frac {K[1]^2+\log (K[1]) K[1]-e^{K[1]}+\log \left (\frac {1}{K[1]}\right )}{K[1] \left (e^{K[1]}-\log \left (\frac {1}{K[1]}\right )\right )}dK[1]\right )}{-\int _1^x-\frac {\exp \left (\int _1^{K[2]}\frac {K[1]^2+\log (K[1]) K[1]-e^{K[1]}+\log \left (\frac {1}{K[1]}\right )}{K[1] \left (e^{K[1]}-\log \left (\frac {1}{K[1]}\right )\right )}dK[1]\right ) K[2] (K[2]+\log (K[2]))}{e^{K[2]}-\log \left (\frac {1}{K[2]}\right )}dK[2]+c_1} \\ y(x)\to 0 \\ y(x)\to -\frac {\exp \left (\int _1^x\frac {K[1]^2+\log (K[1]) K[1]-e^{K[1]}+\log \left (\frac {1}{K[1]}\right )}{K[1] \left (e^{K[1]}-\log \left (\frac {1}{K[1]}\right )\right )}dK[1]\right )}{\int _1^x-\frac {\exp \left (\int _1^{K[2]}\frac {K[1]^2+\log (K[1]) K[1]-e^{K[1]}+\log \left (\frac {1}{K[1]}\right )}{K[1] \left (e^{K[1]}-\log \left (\frac {1}{K[1]}\right )\right )}dK[1]\right ) K[2] (K[2]+\log (K[2]))}{e^{K[2]}-\log \left (\frac {1}{K[2]}\right )}dK[2]} \\ \end{align*}