ODE
\[ \text {a1} (a+b x) y'(x)+(a+b x)^2 y''(x)+\text {a2} y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.177137 (sec), leaf count = 95
\[\left \{\left \{y(x)\to (a+b x)^{-\frac {\sqrt {\text {a2}} \sqrt {\frac {\text {a1}^2-2 \text {a1} b-4 \text {a2}+b^2}{\text {a2}}}+\text {a1}-b}{2 b}} \left (c_2 (a+b x)^{\frac {\sqrt {\text {a2}} \sqrt {\frac {\text {a1}^2-2 \text {a1} b-4 \text {a2}+b^2}{\text {a2}}}}{b}}+c_1\right )\right \}\right \}\]
Maple ✓
cpu = 0.041 (sec), leaf count = 77
\[\left [y \left (x \right ) = \textit {\_C1} \left (x +\frac {a}{b}\right )^{\frac {-\mathit {a1} +b +\sqrt {\mathit {a1}^{2}-2 \mathit {a1} b +b^{2}-4 \mathit {a2}}}{2 b}}+\textit {\_C2} \left (x +\frac {a}{b}\right )^{-\frac {\mathit {a1} -b +\sqrt {\mathit {a1}^{2}-2 \mathit {a1} b +b^{2}-4 \mathit {a2}}}{2 b}}\right ]\] Mathematica raw input
DSolve[a2*y[x] + a1*(a + b*x)*y'[x] + (a + b*x)^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[1] + (a + b*x)^((Sqrt[a2]*Sqrt[(a1^2 - 4*a2 - 2*a1*b + b^2)/a2])/b)*C[2])/(a + b*x)^((a1 - b + Sqrt[a2]*Sqrt[(a1^2 - 4*a2 - 2*a1*b + b^2)/a2])/(2*b))}}
Maple raw input
dsolve((b*x+a)^2*diff(diff(y(x),x),x)+a1*(b*x+a)*diff(y(x),x)+a2*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*(x+a/b)^(1/2*(-a1+b+(a1^2-2*a1*b+b^2-4*a2)^(1/2))/b)+_C2*(x+a/b)^(-1/2*(a1-b+(a1^2-2*a1*b+b^2-4*a2)^(1/2))/b)]