ODE
\[ (a+b x) y'(x)+c y(x)+x y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.178775 (sec), leaf count = 48
\[\left \{\left \{y(x)\to e^{-b x} \left (c_1 U\left (a-\frac {c}{b},a,b x\right )+c_2 L_{\frac {c}{b}-a}^{a-1}(b x)\right )\right \}\right \}\]
Maple ✓
cpu = 0.395 (sec), leaf count = 51
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-b x} \KummerM \left (\frac {a b -c}{b}, a , b x \right )+\textit {\_C2} \,{\mathrm e}^{-b x} \KummerU \left (\frac {a b -c}{b}, a , b x \right )\right ]\] Mathematica raw input
DSolve[c*y[x] + (a + b*x)*y'[x] + x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[1]*HypergeometricU[a - c/b, a, b*x] + C[2]*LaguerreL[-a + c/b, -1 + a, b*x])/E^(b*x)}}
Maple raw input
dsolve(x*diff(diff(y(x),x),x)+(b*x+a)*diff(y(x),x)+c*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*exp(-b*x)*KummerM((a*b-c)/b,a,b*x)+_C2*exp(-b*x)*KummerU((a*b-c)/b,a,b*x)]