4.30.42 $(a+bx)y^{\prime }(x)+cy(x)+x^2{y}^{″}(x)=0$

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

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0728397 (sec), leaf count = 243

$$\left\{\{y(x)\to -i^{-\sqrt{b^2-2b-4c+1}+b+1}{a}^{\frac{1}{2}(-\sqrt{b^2-2b-4c+1}+b-1)}{\left(\frac{1}{x}\right)}^{\frac{1}{2}(-\sqrt{b^2-2b-4c+1}+b-1)}(c_1{}_1{F}_1(\frac{1}{2}(b-\sqrt{b^2-2b-4c+1}-1);1-\sqrt{b^2-2b-4c+1};\frac{a}{x})+c_2{i}^{2\sqrt{b^2-2b-4c+1}}{a}^{\sqrt{b^2-2b-4c+1}}{\left(\frac{1}{x}\right)}^{\sqrt{b^2-2b-4c+1}}{}_1{F}_1(\frac{1}{2}(b+\sqrt{b^2-2b-4c+1}-1);\sqrt{b^2-2b-4c+1}+1;\frac{a}{x}))\}\right\}$$

Maple ✓
cpu = 0.184 (sec), leaf count = 114

$$\{y\left(x\right)=x^{-\frac{1}{2}\sqrt{b^2-2b-4c+1}-\frac{b}{2}+\frac{1}{2}}(\mathit{U}(-\frac{1}{2}+\frac{1}{2}\sqrt{b^2-2b-4c+1}+\frac{b}{2},1+\sqrt{b^2-2b-4c+1},\frac{a}{x})\mathit{\_}\mathit{C}\mathit{2}+\mathit{M}(-\frac{1}{2}+\frac{1}{2}\sqrt{b^2-2b-4c+1}+\frac{b}{2},1+\sqrt{b^2-2b-4c+1},\frac{a}{x})\mathit{\_}\mathit{C}\mathit{1})\}$$ Mathematica raw input

DSolve[c*y[x] + (a + b*x)*y'[x] + x^2*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(I^(1 + b - Sqrt[1 - 2*b + b^2 - 4*c])*a^((-1 + b - Sqrt[1 - 2*b + b^
2 - 4*c])/2)*(x^(-1))^((-1 + b - Sqrt[1 - 2*b + b^2 - 4*c])/2)*(C[1]*Hypergeomet
ric1F1[(-1 + b - Sqrt[1 - 2*b + b^2 - 4*c])/2, 1 - Sqrt[1 - 2*b + b^2 - 4*c], a/
x] + I^(2*Sqrt[1 - 2*b + b^2 - 4*c])*a^Sqrt[1 - 2*b + b^2 - 4*c]*(x^(-1))^Sqrt[1
 - 2*b + b^2 - 4*c]*C[2]*Hypergeometric1F1[(-1 + b + Sqrt[1 - 2*b + b^2 - 4*c])/
2, 1 + Sqrt[1 - 2*b + b^2 - 4*c], a/x]))}}

Maple raw input

dsolve(x^2*diff(diff(y(x),x),x)+(b*x+a)*diff(y(x),x)+c*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = x^(-1/2*(b^2-2*b-4*c+1)^(1/2)-1/2*b+1/2)*(KummerU(-1/2+1/2*(b^2-2*b-4*c+1
)^(1/2)+1/2*b,1+(b^2-2*b-4*c+1)^(1/2),1/x*a)*_C2+KummerM(-1/2+1/2*(b^2-2*b-4*c+1
)^(1/2)+1/2*b,1+(b^2-2*b-4*c+1)^(1/2),1/x*a)*_C1)