4.28.41 $-2(a+bx)y^{\prime }(x)+y(x)(2a+bx)+x{y}^{″}(x)=0$

ODE
$$-2(a+bx)y^{\prime }(x)+y(x)(2a+bx)+x{y}^{″}(x)=0$$ ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0247 (sec), leaf count = 117

$$\left\{\{y(x)\to x^{2a+1}{e}^{bx-\sqrt{b-1}\sqrt{b}x}(c_{1}U(\sqrt{\frac{b-1}{b}}a+a+1,2a+2,2\sqrt{b-1}\sqrt{b}x)+c_2{L}_{-a(\sqrt{\frac{b-1}{b}}+1)-1}^{2a+1}\left(2\sqrt{b-1}\sqrt{b}x\right))\}\right\}$$

Maple ✓
cpu = 0.198 (sec), leaf count = 106

$$\{y\left(x\right)={\mathrm{e}}^{x(b-\sqrt{b}\sqrt{-1+b})}{x}^{2a+1}(\mathit{M}(1(\sqrt{-1+b}a+a\sqrt{b}+\sqrt{b})\frac{1}{\sqrt{b}},2a+2,2\sqrt{b}\sqrt{-1+b}x)\mathit{\_}\mathit{C}\mathit{1}+\mathit{U}(1(\sqrt{-1+b}a+a\sqrt{b}+\sqrt{b})\frac{1}{\sqrt{b}},2a+2,2\sqrt{b}\sqrt{-1+b}x)\mathit{\_}\mathit{C}\mathit{2})\}$$ 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),'implicit')

Maple raw output

y(x) = exp(x*(b-b^(1/2)*(-1+b)^(1/2)))*x^(2*a+1)*(KummerM(((-1+b)^(1/2)*a+a*b^(1
/2)+b^(1/2))/b^(1/2),2*a+2,2*b^(1/2)*(-1+b)^(1/2)*x)*_C1+KummerU(((-1+b)^(1/2)*a
+a*b^(1/2)+b^(1/2))/b^(1/2),2*a+2,2*b^(1/2)*(-1+b)^(1/2)*x)*_C2)