4.34.19 \(x (\text {a1}+\text {b1} x) y'(x)+y(x) (\text {a2}+\text {b2} x)+x^3 y''(x)=0\)

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 = 0.110589 (sec), leaf count = 261

\[\left \{\left \{y(x)\to -i^{-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}+\text {b1}+1} \text {a1}^{\frac {1}{2} \left (-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}+\text {b1}-1\right )} \left (\frac {1}{x}\right )^{\frac {1}{2} \left (-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}+\text {b1}-1\right )} \left (c_1 \, _1F_1\left (-\frac {-\text {b1} \text {a1}+\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1} \text {a1}+\text {a1}+2 \text {a2}}{2 \text {a1}};1-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1};\frac {\text {a1}}{x}\right )+c_2 i^{2 \sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}} \text {a1}^{\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}} \left (\frac {1}{x}\right )^{\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}} \, _1F_1\left (\frac {\text {a1} \left (\text {b1}+\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}-1\right )-2 \text {a2}}{2 \text {a1}};\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}+1;\frac {\text {a1}}{x}\right )\right )\right \}\right \}\]

Maple ✓
cpu = 0.243 (sec), leaf count = 132

\[ \left \{ y \left ( x \right ) ={x}^{-{\frac {{\it b1}}{2}}+{\frac {1}{2}}-{\frac {1}{2}\sqrt {{{\it b1}}^{2}-2\,{\it b1}-4\,{\it b2}+1}}} \left ( {{\sl U}\left ({\frac {1}{2\,{\it a1}} \left ( \sqrt {{{\it b1}}^{2}-2\,{\it b1}-4\,{\it b2}+1}{\it a1}+{\it a1}\, \left ( -1+{\it b1} \right ) -2\,{\it a2} \right ) },\,1+\sqrt {{{\it b1}}^{2}-2\,{\it b1}-4\,{\it b2}+1},\,{\frac {{\it a1}}{x}}\right )}{\it \_C2}+{{\sl M}\left ({\frac {1}{2\,{\it a1}} \left ( \sqrt {{{\it b1}}^{2}-2\,{\it b1}-4\,{\it b2}+1}{\it a1}+{\it a1}\, \left ( -1+{\it b1} \right ) -2\,{\it a2} \right ) },\,1+\sqrt {{{\it b1}}^{2}-2\,{\it b1}-4\,{\it b2}+1},\,{\frac {{\it a1}}{x}}\right )}{\it \_C1} \right ) \right \} \] 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)*(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[(-2*a2 + a1*(-1 + b1 + Sqrt[1 - 2*b
1 + b1^2 - 4*b2]))/(2*a1), 1 + Sqrt[1 - 2*b1 + b1^2 - 4*b2], a1/x] + C[1]*Hyperg
eometric1F1[-(a1 + 2*a2 - a1*b1 + a1*Sqrt[1 - 2*b1 + b1^2 - 4*b2])/(2*a1), 1 - S
qrt[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),'implicit')

Maple raw output

y(x) = 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*b
2+1)^(1/2)*a1+a1*(-1+b1)-2*a2)/a1,1+(b1^2-2*b1-4*b2+1)^(1/2),1/x*a1)*_C2+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)*_C1)