ODE
\[ -2 (a+b x) y'(x)+y(x) (2 a+b x)+x y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.209242 (sec), leaf count = 117
\[\left \{\left \{y(x)\to x^{2 a+1} e^{b x-\sqrt {b-1} \sqrt {b} x} \left (c_1 U\left (\sqrt {\frac {b-1}{b}} a+a+1,2 a+2,2 \sqrt {b-1} \sqrt {b} x\right )+c_2 L_{-a \left (\sqrt {\frac {b-1}{b}}+1\right )-1}^{2 a+1}\left (2 \sqrt {b-1} \sqrt {b} x\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.53 (sec), leaf count = 127
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{x \left (-\sqrt {b}\, \sqrt {b -1}+b \right )} \KummerM \left (\frac {\sqrt {b}\, a +a \sqrt {b -1}+\sqrt {b}}{\sqrt {b}}, 2+2 a , 2 \sqrt {b}\, \sqrt {b -1}\, x \right ) x^{2 a +1}+\textit {\_C2} \,{\mathrm e}^{x \left (-\sqrt {b}\, \sqrt {b -1}+b \right )} \KummerU \left (\frac {\sqrt {b}\, a +a \sqrt {b -1}+\sqrt {b}}{\sqrt {b}}, 2+2 a , 2 \sqrt {b}\, \sqrt {b -1}\, x \right ) x^{2 a +1}\right ]\] Mathematica raw input
DSolve[(2*a + b*x)*y[x] - 2*(a + b*x)*y'[x] + x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> E^(-(Sqrt[-1 + b]*Sqrt[b]*x) + b*x)*x^(1 + 2*a)*(C[1]*HypergeometricU[1 + a + a*Sqrt[(-1 + b)/b], 2 + 2*a, 2*Sqrt[-1 + b]*Sqrt[b]*x] + C[2]*LaguerreL[-1 - a*(1 + Sqrt[(-1 + b)/b]), 1 + 2*a, 2*Sqrt[-1 + b]*Sqrt[b]*x])}}
Maple raw input
dsolve(x*diff(diff(y(x),x),x)-2*(b*x+a)*diff(y(x),x)+(b*x+2*a)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*exp(x*(-b^(1/2)*(b-1)^(1/2)+b))*KummerM((b^(1/2)*a+a*(b-1)^(1/2)+b^(1/2))/b^(1/2),2+2*a,2*b^(1/2)*(b-1)^(1/2)*x)*x^(2*a+1)+_C2*exp(x*(-b^(1/2)*(b-1)^(1/2)+b))*KummerU((b^(1/2)*a+a*(b-1)^(1/2)+b^(1/2))/b^(1/2),2+2*a,2*b^(1/2)*(b-1)^(1/2)*x)*x^(2*a+1)]