ODE
\[ (\text {a1}+\text {b1} x) y'(x)+y(x) (\text {a2}+\text {b2} x)+x y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0221795 (sec), leaf count = 134
\[\left \{\left \{y(x)\to e^{-\frac {1}{2} x \left (\sqrt {\text {b1}^2-4 \text {b2}}+\text {b1}\right )} \left (c_1 U\left (\frac {\text {a1} \left (\text {b1}+\sqrt {\text {b1}^2-4 \text {b2}}\right )-2 \text {a2}}{2 \sqrt {\text {b1}^2-4 \text {b2}}},\text {a1},\sqrt {\text {b1}^2-4 \text {b2}} x\right )+c_2 L_{\frac {2 \text {a2}-\text {a1} \left (\sqrt {\text {b1}^2-4 \text {b2}}+\text {b1}\right )}{2 \sqrt {\text {b1}^2-4 \text {b2}}}}^{\text {a1}-1}\left (x \sqrt {\text {b1}^2-4 \text {b2}}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.253 (sec), leaf count = 109
\[ \left \{ y \left ( x \right ) ={{\rm e}^{-{\frac {x}{2} \left ( {\it b1}+\sqrt {{{\it b1}}^{2}-4\,{\it b2}} \right ) }}} \left ( {{\sl U}\left ({\frac {1}{2} \left ( {\it a1}\,\sqrt {{{\it b1}}^{2}-4\,{\it b2}}+{\it a1}\,{\it b1}-2\,{\it a2} \right ) {\frac {1}{\sqrt {{{\it b1}}^{2}-4\,{\it b2}}}}},\,{\it a1},\,\sqrt {{{\it b1}}^{2}-4\,{\it b2}}x\right )}{\it \_C2}+{{\sl M}\left ({\frac {1}{2} \left ( {\it a1}\,\sqrt {{{\it b1}}^{2}-4\,{\it b2}}+{\it a1}\,{\it b1}-2\,{\it a2} \right ) {\frac {1}{\sqrt {{{\it b1}}^{2}-4\,{\it b2}}}}},\,{\it a1},\,\sqrt {{{\it b1}}^{2}-4\,{\it b2}}x\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[(a2 + b2*x)*y[x] + (a1 + b1*x)*y'[x] + x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[1]*HypergeometricU[(-2*a2 + a1*(b1 + Sqrt[b1^2 - 4*b2]))/(2*Sqrt[b1^2 - 4*b2]), a1, Sqrt[b1^2 - 4*b2]*x] + C[2]*LaguerreL[(2*a2 - a1*(b1 + Sqrt[b1^2 - 4*b2]))/(2*Sqrt[b1^2 - 4*b2]), -1 + a1, Sqrt[b1^2 - 4*b2]*x])/E^(((b1 + Sqrt[b1^2 - 4*b2])*x)/2)}}
Maple raw input
dsolve(x*diff(diff(y(x),x),x)+(b1*x+a1)*diff(y(x),x)+(b2*x+a2)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = exp(-1/2*x*(b1+(b1^2-4*b2)^(1/2)))*(KummerU(1/2*(a1*(b1^2-4*b2)^(1/2)+a1*b1-2*a2)/(b1^2-4*b2)^(1/2),a1,(b1^2-4*b2)^(1/2)*x)*_C2+KummerM(1/2*(a1*(b1^2-4*b2)^(1/2)+a1*b1-2*a2)/(b1^2-4*b2)^(1/2),a1,(b1^2-4*b2)^(1/2)*x)*_C1)