2.1041   ODE No. 1041

\[ (n+1) y(x)+y''(x)+x y'(x)=0 \]

✓ Mathematica : cpu = 0.0063327 (sec), leaf count = 55

DSolve[(1 + n)*y[x] + x*Derivative[1][y][x] + Derivative[2][y][x] == 0,y[x],x]
 

\[\left \{\left \{y(x)\to c_1 e^{-\frac {x^2}{2}} H_n\left (\frac {x}{\sqrt {2}}\right )+c_2 e^{-\frac {x^2}{2}} \, _1F_1\left (-\frac {n}{2};\frac {1}{2};\frac {x^2}{2}\right )\right \}\right \}\]

✓ Maple : cpu = 0.13 (sec), leaf count = 41

dsolve(diff(diff(y(x),x),x)+x*diff(y(x),x)+(n+1)*y(x)=0,y(x))
 

\[y \left (x \right ) = {\mathrm e}^{-\frac {x^{2}}{2}} x \left (\operatorname {KummerU}\left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right ) c_{2} +\operatorname {KummerM}\left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right ) c_{1} \right )\]