60.2.138 problem 714
Internal
problem
ID
[10725]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
714
Date
solved
:
Monday, January 27, 2025 at 09:34:45 PM
CAS
classification
:
[_Bernoulli]
\begin{align*} y^{\prime }&=-\frac {y \left (-\ln \left (\frac {1}{x}\right )+{\mathrm e}^{x}+y x^{2} \ln \left (x \right )+x^{3} y-x \ln \left (x \right )-x^{2}\right )}{\left (-\ln \left (\frac {1}{x}\right )+{\mathrm e}^{x}\right ) x} \end{align*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 102
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)
\[
y = \frac {{\mathrm e}^{-\int \frac {\ln \left (x \right ) x +x^{2}+\ln \left (\frac {1}{x}\right )-{\mathrm e}^{x}}{x \left (\ln \left (\frac {1}{x}\right )-{\mathrm e}^{x}\right )}d x}}{-\int \frac {{\mathrm e}^{-\int \frac {\ln \left (x \right ) x +x^{2}+\ln \left (\frac {1}{x}\right )-{\mathrm e}^{x}}{x \left (\ln \left (\frac {1}{x}\right )-{\mathrm e}^{x}\right )}d x} x \left (x +\ln \left (x \right )\right )}{\ln \left (\frac {1}{x}\right )-{\mathrm e}^{x}}d x +c_{1}}
\]
✓ Solution by Mathematica
Time used: 1.223 (sec). Leaf size: 290
DSolve[D[y[x],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*}