Internal problem ID [8536]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 956.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y \left (-1-x^{\frac {2}{\ln \relax (x )+1}} {\mathrm e}^{\frac {2 \ln \relax (x )^{2}}{\ln \relax (x )+1}} x^{2}-x^{\frac {2}{\ln \relax (x )+1}} {\mathrm e}^{\frac {2 \ln \relax (x )^{2}}{\ln \relax (x )+1}} x^{2} \ln \relax (x )+x^{\frac {2}{\ln \relax (x )+1}} {\mathrm e}^{\frac {2 \ln \relax (x )^{2}}{\ln \relax (x )+1}} x^{2} y+2 x^{\frac {2}{\ln \relax (x )+1}} {\mathrm e}^{\frac {2 \ln \relax (x )^{2}}{\ln \relax (x )+1}} x^{2} y \ln \relax (x )+x^{\frac {2}{\ln \relax (x )+1}} {\mathrm e}^{\frac {2 \ln \relax (x )^{2}}{\ln \relax (x )+1}} x^{2} y \ln \relax (x )^{2}\right )}{\left (\ln \relax (x )+1\right ) x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 188
dsolve(diff(y(x),x) = 1/(1+ln(x))*y(x)*(-1-x^(2/(1+ln(x)))*exp(2/(1+ln(x))*ln(x)^2)*x^2-x^(2/(1+ln(x)))*exp(2/(1+ln(x))*ln(x)^2)*x^2*ln(x)+x^(2/(1+ln(x)))*exp(2/(1+ln(x))*ln(x)^2)*x^2*y(x)+2*x^(2/(1+ln(x)))*exp(2/(1+ln(x))*ln(x)^2)*x^2*y(x)*ln(x)+x^(2/(1+ln(x)))*exp(2/(1+ln(x))*ln(x)^2)*x^2*y(x)*ln(x)^2)/x,y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{-\frac {x^{4}}{4}}}{{\mathrm e}^{-\frac {x^{4} \ln \relax (x )+x^{4}+4 \ln \left (\ln \relax (x )+1\right ) \ln \relax (x )-8 \ln \relax (x )^{2}+4 \ln \left (\ln \relax (x )+1\right )}{4 \left (\ln \relax (x )+1\right )}} x^{-\frac {2 \ln \relax (x )}{\ln \relax (x )+1}} \ln \relax (x )^{2}+2 \,{\mathrm e}^{-\frac {x^{4} \ln \relax (x )+x^{4}+4 \ln \left (\ln \relax (x )+1\right ) \ln \relax (x )-8 \ln \relax (x )^{2}+4 \ln \left (\ln \relax (x )+1\right )}{4 \left (\ln \relax (x )+1\right )}} x^{-\frac {2 \ln \relax (x )}{\ln \relax (x )+1}} \ln \relax (x )+{\mathrm e}^{-\frac {x^{4} \ln \relax (x )+x^{4}+4 \ln \left (\ln \relax (x )+1\right ) \ln \relax (x )-8 \ln \relax (x )^{2}+4 \ln \left (\ln \relax (x )+1\right )}{4 \left (\ln \relax (x )+1\right )}} x^{-\frac {2 \ln \relax (x )}{\ln \relax (x )+1}}+c_{1} \ln \relax (x )+c_{1}} \]
✓ Solution by Mathematica
Time used: 1.546 (sec). Leaf size: 32
DSolve[y'[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])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{c_1 e^{\frac {x^4}{4}} (\log (x)+1)+\log (x)+1} \\ y(x)\to 0 \\ \end{align*}