2.612   ODE No. 612

\[ y'(x)=\frac {1}{2} e^{\frac {x^2}{4}} \left (2 F\left (e^{-\frac {x^2}{4}} y(x)\right )+e^{-\frac {x^2}{4}} x y(x)\right ) \]

✓ Mathematica : cpu = 0.34395 (sec), leaf count = 199

DSolve[Derivative[1][y][x] == (E^(x^2/4)*(2*F[y[x]/E^(x^2/4)] + (x*y[x])/E^(x^2/4)))/2,y[x],x]
 

\[\text {Solve}\left [\int _1^{y(x)}-\frac {e^{-\frac {x^2}{4}} \left (e^{\frac {x^2}{4}} F\left (e^{-\frac {x^2}{4}} K[2]\right ) \int _1^x\left (\frac {e^{-\frac {1}{4} K[1]^2} K[1]}{2 F\left (e^{-\frac {1}{4} K[1]^2} K[2]\right )}-\frac {e^{-\frac {1}{2} K[1]^2} K[1] K[2] F'\left (e^{-\frac {1}{4} K[1]^2} K[2]\right )}{2 F\left (e^{-\frac {1}{4} K[1]^2} K[2]\right )^2}\right )dK[1]+1\right )}{F\left (e^{-\frac {x^2}{4}} K[2]\right )}dK[2]+\int _1^x\left (\frac {e^{-\frac {1}{4} K[1]^2} K[1] y(x)}{2 F\left (e^{-\frac {1}{4} K[1]^2} y(x)\right )}+1\right )dK[1]=c_1,y(x)\right ]\]

✓ Maple : cpu = 0.102 (sec), leaf count = 27

dsolve(diff(y(x),x) = 1/2*(y(x)*exp(-1/4*x^2)*x+2*F(y(x)*exp(-1/4*x^2)))*exp(1/4*x^2),y(x))
 

\[y \left (x \right ) = \operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right ) {\mathrm e}^{\frac {x^{2}}{4}}\]