60.2.143 problem 719

Internal problem ID [10730]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 719
Date solved : Monday, January 27, 2025 at 09:35:42 PM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 36

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

\[ y = \frac {1}{x \left (1+2^{-{\mathrm e}^{-x}} x^{-{\mathrm e}^{-x}} {\mathrm e}^{-\operatorname {Ei}_{1}\left (x \right )} c_{1} \right )} \]

✓ Solution by Mathematica

Time used: 0.855 (sec). Leaf size: 184

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

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