Internal problem ID [8381]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 801.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Abel]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (y \,{\mathrm e}^{-\frac {x^{2}}{4}} x +2+2 y^{2} {\mathrm e}^{-\frac {x^{2}}{2}}+2 y^{3} {\mathrm e}^{-\frac {3 x^{2}}{4}}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 45
dsolve(diff(y(x),x) = 1/2*(y(x)*exp(-1/4*x^2)*x+2+2*y(x)^2*exp(-1/2*x^2)+2*y(x)^3*exp(-3/4*x^2))*exp(1/4*x^2),y(x), singsol=all)
\[ y \relax (x ) = \frac {29 \,{\mathrm e}^{\frac {x^{2}}{4}} \RootOf \left (-81 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+x +3 c_{1}\right )}{9}-\frac {{\mathrm e}^{\frac {x^{2}}{4}}}{3} \]
✓ Solution by Mathematica
Time used: 0.261 (sec). Leaf size: 126
DSolve[y'[x] == (E^(x^2/4)*(2 + (x*y[x])/E^(x^2/4) + (2*y[x]^2)/E^(x^2/2) + (2*y[x]^3)/E^((3*x^2)/4)))/2,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {29}{3} \text {RootSum}\left [-29 \text {$\#$1}^3+3 \sqrt [3]{29} \text {$\#$1}-29\&,\frac {\log \left (\frac {3 e^{-\frac {x^2}{2}} y(x)+e^{-\frac {x^2}{4}}}{\sqrt [3]{29} \sqrt [3]{e^{-\frac {3 x^2}{4}}}}-\text {$\#$1}\right )}{\sqrt [3]{29}-29 \text {$\#$1}^2}\&\right ]=\frac {1}{9} 29^{2/3} e^{\frac {x^2}{2}} \left (e^{-\frac {3 x^2}{4}}\right )^{2/3} x+c_1,y(x)\right ] \]