60.2.378 problem 956

Internal problem ID [10952]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 956
Date solved : Sunday, March 30, 2025 at 07:30:28 PM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }&=\frac {y \left (-1-x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2}-x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} \ln \left (x \right )+x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} y+2 x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} y \ln \left (x \right )+x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} y \ln \left (x \right )^{2}\right )}{\left (1+\ln \left (x \right )\right ) x} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 23

ode:=diff(y(x),x) = 1/(ln(x)+1)*y(x)*(-1-x^(2/(ln(x)+1))*exp(2/(ln(x)+1)*ln(x)^2)*x^2-x^(2/(ln(x)+1))*exp(2/(ln(x)+1)*ln(x)^2)*x^2*ln(x)+x^(2/(ln(x)+1))*exp(2/(ln(x)+1)*ln(x)^2)*x^2*y(x)+2*x^(2/(ln(x)+1))*exp(2/(ln(x)+1)*ln(x)^2)*x^2*y(x)*ln(x)+x^(2/(ln(x)+1))*exp(2/(ln(x)+1)*ln(x)^2)*x^2*y(x)*ln(x)^2)/x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {1}{\left (1+\ln \left (x \right )\right ) \left ({\mathrm e}^{\frac {x^{4}}{4}} c_1 +1\right )} \]

✓ Mathematica. Time used: 2.002 (sec). Leaf size: 452

ode=D[y[x],x] == (y[x]*(-1 - E^((2*Log[x]^2)/(1 + Log[x]))*x^(2 + 2/(1 + Log[x])) - E^((2*Log[x]^2)/(1 + Log[x]))*x^(2 + 2/(1 + Log[x]))*Log[x] + E^((2*Log[x]^2)/(1 + Log[x]))*x^(2 + 2/(1 + Log[x]))*y[x] + 2*E^((2*Log[x]^2)/(1 + Log[x]))*x^(2 + 2/(1 + Log[x]))*Log[x]*y[x] + E^((2*Log[x]^2)/(1 + Log[x]))*x^(2 + 2/(1 + Log[x]))*Log[x]^2*y[x]))/(x*(1 + Log[x])); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {\exp \left (\int _1^x-\frac {e^{\frac {4 \log ^2(K[1])}{\log (K[1])+1}} K[1]^{\frac {4}{\log (K[1])+1}}+e^{\frac {4 \log ^2(K[1])}{\log (K[1])+1}} \log (K[1]) K[1]^{\frac {4}{\log (K[1])+1}}+1}{\log (K[1]) K[1]+K[1]}dK[1]\right )}{-\int _1^x\exp \left (\frac {2 \log ^2(K[2])}{\log (K[2])+1}+\int _1^{K[2]}-\frac {e^{\frac {4 \log ^2(K[1])}{\log (K[1])+1}} K[1]^{\frac {4}{\log (K[1])+1}}+e^{\frac {4 \log ^2(K[1])}{\log (K[1])+1}} \log (K[1]) K[1]^{\frac {4}{\log (K[1])+1}}+1}{\log (K[1]) K[1]+K[1]}dK[1]\right ) K[2]^{1+\frac {2}{\log (K[2])+1}} (\log (K[2])+1)dK[2]+c_1} \\ y(x)\to 0 \\ y(x)\to -\frac {\exp \left (\int _1^x-\frac {e^{\frac {4 \log ^2(K[1])}{\log (K[1])+1}} K[1]^{\frac {4}{\log (K[1])+1}}+e^{\frac {4 \log ^2(K[1])}{\log (K[1])+1}} \log (K[1]) K[1]^{\frac {4}{\log (K[1])+1}}+1}{\log (K[1]) K[1]+K[1]}dK[1]\right )}{\int _1^x\exp \left (\frac {2 \log ^2(K[2])}{\log (K[2])+1}+\int _1^{K[2]}-\frac {e^{\frac {4 \log ^2(K[1])}{\log (K[1])+1}} K[1]^{\frac {4}{\log (K[1])+1}}+e^{\frac {4 \log ^2(K[1])}{\log (K[1])+1}} \log (K[1]) K[1]^{\frac {4}{\log (K[1])+1}}+1}{\log (K[1]) K[1]+K[1]}dK[1]\right ) K[2]^{1+\frac {2}{\log (K[2])+1}} (\log (K[2])+1)dK[2]} \\ \end{align*}

✓ Sympy. Time used: 1.346 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**2*x**(2/(log(x) + 1))*y(x)*exp(2*log(x)**2/(log(x) + 1))*log(x)**2 + 2*x**2*x**(2/(log(x) + 1))*y(x)*exp(2*log(x)**2/(log(x) + 1))*log(x) + x**2*x**(2/(log(x) + 1))*y(x)*exp(2*log(x)**2/(log(x) + 1)) - x**2*x**(2/(log(x) + 1))*exp(2*log(x)**2/(log(x) + 1))*log(x) - x**2*x**(2/(log(x) + 1))*exp(2*log(x)**2/(log(x) + 1)) - 1)*y(x)/(x*(log(x) + 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {1}{C_{1} e^{\frac {x^{4}}{4}} \log {\left (x \right )} + C_{1} e^{\frac {x^{4}}{4}} + \log {\left (x \right )} + 1} \]