60.2.143 problem 719

Internal problem ID [10717]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 719
Date solved : Sunday, March 30, 2025 at 06:26:17 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*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 36

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

\[ y = \frac {1}{x \left (1+c_1 2^{-{\mathrm e}^{-x}} x^{-{\mathrm e}^{-x}} {\mathrm e}^{-\operatorname {Ei}_{1}\left (x \right )}\right )} \]

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

ode=D[y[x],x] == (y[x]*(-E^x - x*Log[2*x] + x^2*Log[2*x]*y[x]))/(E^x*x); 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 44.002 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**2*y(x)*log(2*x) - x*log(2*x) - exp(x))*y(x)*exp(-x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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