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 = 0.0197895 (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 = 0.096 (sec), leaf count = 58
\[ \left \{ y \left ( x \right ) ={{\rm e}^{-{\frac {a{x}^{2}}{2}}}}x \left ( {{\sl U}\left ({\frac {2\,a-b}{2\,a}},\,{\frac {3}{2}},\,{\frac {a{x}^{2}}{2}}\right )}{\it \_C2}+{{\sl M}\left ({\frac {2\,a-b}{2\,a}},\,{\frac {3}{2}},\,{\frac {a{x}^{2}}{2}}\right )}{\it \_C1} \right ) \right \} \] 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),'implicit')
Maple raw output
y(x) = exp(-1/2*a*x^2)*x*(KummerU(1/2/a*(2*a-b),3/2,1/2*a*x^2)*_C2+KummerM(1/2/a*(2*a-b),3/2,1/2*a*x^2)*_C1)