ODE
\[ a x y'(x)+b y(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.167088 (sec), leaf count = 67
\[\left \{\left \{y(x)\to e^{-\frac {a x^2}{2}} \left (c_1 H_{\frac {b}{a}-1}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )+c_2 \, _1F_1\left (\frac {a-b}{2 a};\frac {1}{2};\frac {a x^2}{2}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.462 (sec), leaf count = 65
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-\frac {a \,x^{2}}{2}} \KummerM \left (\frac {2 a -b}{2 a}, \frac {3}{2}, \frac {a \,x^{2}}{2}\right ) x +\textit {\_C2} \,{\mathrm e}^{-\frac {a \,x^{2}}{2}} \KummerU \left (\frac {2 a -b}{2 a}, \frac {3}{2}, \frac {a \,x^{2}}{2}\right ) x\right ]\] Mathematica raw input
DSolve[b*y[x] + a*x*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[1]*HermiteH[-1 + b/a, (Sqrt[a]*x)/Sqrt[2]] + C[2]*Hypergeometric1F1[(a - b)/(2*a), 1/2, (a*x^2)/2])/E^((a*x^2)/2)}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+a*x*diff(y(x),x)+b*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*exp(-1/2*a*x^2)*KummerM(1/2/a*(2*a-b),3/2,1/2*a*x^2)*x+_C2*exp(-1/2*a*x^2)*KummerU(1/2/a*(2*a-b),3/2,1/2*a*x^2)*x]