60.2.380 problem 957
Internal
problem
ID
[10967]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
957
Date
solved
:
Tuesday, January 28, 2025 at 05:39:00 PM
CAS
classification
:
[_Bernoulli]
\begin{align*} y^{\prime }&=\frac {y \left (-1-x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}}-x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} \ln \left (x \right )+x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} y+2 x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} y \ln \left (x \right )+x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} y \ln \left (x \right )^{2}\right )}{\left (1+\ln \left (x \right )\right ) x} \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 27
dsolve(diff(y(x),x) = 1/(1+ln(x))*y(x)*(-1-x^3*x^(2/(1+ln(x)))*exp(2/(1+ln(x))*ln(x)^2)-x^3*x^(2/(1+ln(x)))*exp(2/(1+ln(x))*ln(x)^2)*ln(x)+x^3*x^(2/(1+ln(x)))*exp(2/(1+ln(x))*ln(x)^2)*y(x)+2*x^3*x^(2/(1+ln(x)))*exp(2/(1+ln(x))*ln(x)^2)*y(x)*ln(x)+x^3*x^(2/(1+ln(x)))*exp(2/(1+ln(x))*ln(x)^2)*y(x)*ln(x)^2)/x,y(x), singsol=all)
\[
y = \frac {{\mathrm e}^{-\frac {x^{5}}{5}}}{\left (1+\ln \left (x \right )\right ) \left ({\mathrm e}^{-\frac {x^{5}}{5}}+c_{1} \right )}
\]
✓ Solution by Mathematica
Time used: 1.991 (sec). Leaf size: 452
DSolve[D[y[x],x] == (y[x]*(-1 - E^((2*Log[x]^2)/(1 + Log[x]))*x^(3 + 2/(1 + Log[x])) - E^((2*Log[x]^2)/(1 + Log[x]))*x^(3 + 2/(1 + Log[x]))*Log[x] + E^((2*Log[x]^2)/(1 + Log[x]))*x^(3 + 2/(1 + Log[x]))*y[x] + 2*E^((2*Log[x]^2)/(1 + Log[x]))*x^(3 + 2/(1 + Log[x]))*Log[x]*y[x] + E^((2*Log[x]^2)/(1 + Log[x]))*x^(3 + 2/(1 + Log[x]))*Log[x]^2*y[x]))/(x*(1 + Log[x])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\exp \left (\int _1^x-\frac {e^{\frac {5 \log ^2(K[1])}{\log (K[1])+1}} K[1]^{\frac {5}{\log (K[1])+1}}+e^{\frac {5 \log ^2(K[1])}{\log (K[1])+1}} \log (K[1]) K[1]^{\frac {5}{\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 {5 \log ^2(K[1])}{\log (K[1])+1}} K[1]^{\frac {5}{\log (K[1])+1}}+e^{\frac {5 \log ^2(K[1])}{\log (K[1])+1}} \log (K[1]) K[1]^{\frac {5}{\log (K[1])+1}}+1}{\log (K[1]) K[1]+K[1]}dK[1]\right ) K[2]^{2+\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 {5 \log ^2(K[1])}{\log (K[1])+1}} K[1]^{\frac {5}{\log (K[1])+1}}+e^{\frac {5 \log ^2(K[1])}{\log (K[1])+1}} \log (K[1]) K[1]^{\frac {5}{\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 {5 \log ^2(K[1])}{\log (K[1])+1}} K[1]^{\frac {5}{\log (K[1])+1}}+e^{\frac {5 \log ^2(K[1])}{\log (K[1])+1}} \log (K[1]) K[1]^{\frac {5}{\log (K[1])+1}}+1}{\log (K[1]) K[1]+K[1]}dK[1]\right ) K[2]^{2+\frac {2}{\log (K[2])+1}} (\log (K[2])+1)dK[2]} \\
\end{align*}