ODE
\[ x (\text {a1}+\text {b1} x) y'(x)+y(x) (\text {a2}+\text {b2} x)+x^3 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.350446 (sec), leaf count = 255
\[\left \{\left \{y(x)\to -i^{-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}+\text {b1}+1} \text {a1}^{\frac {1}{2} \left (-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}+\text {b1}-1\right )} \left (\frac {1}{x}\right )^{\frac {1}{2} \left (-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}+\text {b1}-1\right )} \left (c_2 i^{2 \sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}} \text {a1}^{\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}} \left (\frac {1}{x}\right )^{\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}} \, _1F_1\left (\frac {1}{2} \left (-\frac {2 \text {a2}}{\text {a1}}+\text {b1}+\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}-1\right );\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}+1;\frac {\text {a1}}{x}\right )+c_1 \, _1F_1\left (\frac {1}{2} \left (-\frac {2 \text {a2}}{\text {a1}}+\text {b1}-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}-1\right );1-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1};\frac {\text {a1}}{x}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.648 (sec), leaf count = 153
\[\left [y \left (x \right ) = \textit {\_C1} \,x^{-\frac {\mathit {b1}}{2}+\frac {1}{2}-\frac {\sqrt {\mathit {b1}^{2}-2 \mathit {b1} -4 \mathit {b2} +1}}{2}} \KummerM \left (\frac {\sqrt {\mathit {b1}^{2}-2 \mathit {b1} -4 \mathit {b2} +1}\, \mathit {a1} +\mathit {a1} \left (-1+\mathit {b1} \right )-2 \mathit {a2}}{2 \mathit {a1}}, 1+\sqrt {\mathit {b1}^{2}-2 \mathit {b1} -4 \mathit {b2} +1}, \frac {\mathit {a1}}{x}\right )+\textit {\_C2} \,x^{-\frac {\mathit {b1}}{2}+\frac {1}{2}-\frac {\sqrt {\mathit {b1}^{2}-2 \mathit {b1} -4 \mathit {b2} +1}}{2}} \KummerU \left (\frac {\sqrt {\mathit {b1}^{2}-2 \mathit {b1} -4 \mathit {b2} +1}\, \mathit {a1} +\mathit {a1} \left (-1+\mathit {b1} \right )-2 \mathit {a2}}{2 \mathit {a1}}, 1+\sqrt {\mathit {b1}^{2}-2 \mathit {b1} -4 \mathit {b2} +1}, \frac {\mathit {a1}}{x}\right )\right ]\] Mathematica raw input
DSolve[(a2 + b2*x)*y[x] + x*(a1 + b1*x)*y'[x] + x^3*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(I^(1 + b1 - Sqrt[1 - 2*b1 + b1^2 - 4*b2])*a1^((-1 + b1 - Sqrt[1 - 2*b1 + b1^2 - 4*b2])/2)*(x^(-1))^((-1 + b1 - Sqrt[1 - 2*b1 + b1^2 - 4*b2])/2)*(C[1]*Hypergeometric1F1[(-1 - (2*a2)/a1 + b1 - Sqrt[1 - 2*b1 + b1^2 - 4*b2])/2, 1 - Sqrt[1 - 2*b1 + b1^2 - 4*b2], a1/x] + I^(2*Sqrt[1 - 2*b1 + b1^2 - 4*b2])*a1^Sqrt[1 - 2*b1 + b1^2 - 4*b2]*(x^(-1))^Sqrt[1 - 2*b1 + b1^2 - 4*b2]*C[2]*Hypergeometric1F1[(-1 - (2*a2)/a1 + b1 + Sqrt[1 - 2*b1 + b1^2 - 4*b2])/2, 1 + Sqrt[1 - 2*b1 + b1^2 - 4*b2], a1/x]))}}
Maple raw input
dsolve(x^3*diff(diff(y(x),x),x)+x*(b1*x+a1)*diff(y(x),x)+(b2*x+a2)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*x^(-1/2*b1+1/2-1/2*(b1^2-2*b1-4*b2+1)^(1/2))*KummerM(1/2*((b1^2-2*b1-4*b2+1)^(1/2)*a1+a1*(-1+b1)-2*a2)/a1,1+(b1^2-2*b1-4*b2+1)^(1/2),1/x*a1)+_C2*x^(-1/2*b1+1/2-1/2*(b1^2-2*b1-4*b2+1)^(1/2))*KummerU(1/2*((b1^2-2*b1-4*b2+1)^(1/2)*a1+a1*(-1+b1)-2*a2)/a1,1+(b1^2-2*b1-4*b2+1)^(1/2),1/x*a1)]