Internal problem ID [8562]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 982.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Abel]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y \,{\mathrm e}^{-\frac {x^{2}}{2}} \left (2 y^{2}+2 y \,{\mathrm e}^{\frac {x^{2}}{4}}+2 \,{\mathrm e}^{\frac {x^{2}}{2}}+x \,{\mathrm e}^{\frac {x^{2}}{2}}\right )}{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 187
dsolve(diff(y(x),x) = 1/2*y(x)/exp(1/4*x^2)^2*(2*y(x)^2+2*y(x)*exp(1/4*x^2)+2*exp(1/4*x^2)^2+x*exp(1/4*x^2)^2),y(x), singsol=all)
\[ -\frac {2 \ln \left (18 y \relax (x ) {\mathrm e}^{-\frac {x^{2}}{2}} {\mathrm e}^{\frac {x^{2}}{4}}+6 \,{\mathrm e}^{-\frac {x^{2}}{4}} {\mathrm e}^{\frac {x^{2}}{4}}-6\right )}{3}+\frac {\ln \left (\frac {324 y \relax (x )^{2} {\mathrm e}^{-x^{2}} {\mathrm e}^{\frac {x^{2}}{2}}}{7}+\frac {216 y \relax (x ) {\mathrm e}^{-\frac {x^{2}}{2}} {\mathrm e}^{\frac {x^{2}}{2}} {\mathrm e}^{-\frac {x^{2}}{4}}}{7}+\frac {36 \,{\mathrm e}^{-\frac {x^{2}}{2}} {\mathrm e}^{\frac {x^{2}}{2}}}{7}+\frac {108 y \relax (x ) {\mathrm e}^{-\frac {x^{2}}{2}} {\mathrm e}^{\frac {x^{2}}{4}}}{7}+\frac {36 \,{\mathrm e}^{-\frac {x^{2}}{4}} {\mathrm e}^{\frac {x^{2}}{4}}}{7}+36\right )}{3}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (6 y \relax (x ) {\mathrm e}^{-\frac {x^{2}}{2}} {\mathrm e}^{\frac {x^{2}}{4}}+2 \,{\mathrm e}^{-\frac {x^{2}}{4}} {\mathrm e}^{\frac {x^{2}}{4}}+1\right ) \sqrt {3}}{9}\right )}{9}+\frac {2 x}{3}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.273 (sec). Leaf size: 132
DSolve[y'[x] == (y[x]*(2*E^(x^2/2) + E^(x^2/2)*x + 2*E^(x^2/4)*y[x] + 2*y[x]^2))/(2*E^(x^2/2)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {7}{3} \text {RootSum}\left [-7 \text {$\#$1}^3+6 \sqrt [3]{-7} \text {$\#$1}-7\&,\frac {\log \left (\frac {3 e^{-\frac {x^2}{2}} y(x)+e^{-\frac {x^2}{4}}}{\sqrt [3]{7} \sqrt [3]{-e^{-\frac {3 x^2}{4}}}}-\text {$\#$1}\right )}{2 \sqrt [3]{-7}-7 \text {$\#$1}^2}\&\right ]=\frac {1}{9} 7^{2/3} e^{\frac {x^2}{2}} \left (-e^{-\frac {3 x^2}{4}}\right )^{2/3} x+c_1,y(x)\right ] \]