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 = 0.189509 (sec), leaf count = 134
\[\left \{\left \{y(x)\to e^{-\frac {1}{2} x \left (\sqrt {\text {b1}^2-4 \text {b2}}+\text {b1}\right )} \left (c_1 U\left (\frac {\text {a1} \left (\text {b1}+\sqrt {\text {b1}^2-4 \text {b2}}\right )-2 \text {a2}}{2 \sqrt {\text {b1}^2-4 \text {b2}}},\text {a1},\sqrt {\text {b1}^2-4 \text {b2}} x\right )+c_2 L_{\frac {2 \text {a2}-\text {a1} \left (\text {b1}+\sqrt {\text {b1}^2-4 \text {b2}}\right )}{2 \sqrt {\text {b1}^2-4 \text {b2}}}}^{\text {a1}-1}\left (\sqrt {\text {b1}^2-4 \text {b2}} x\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.644 (sec), leaf count = 123
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-\frac {x \left (\mathit {b1} +\sqrt {\mathit {b1}^{2}-4 \mathit {b2}}\right )}{2}} \KummerM \left (\frac {\mathit {a1} \sqrt {\mathit {b1}^{2}-4 \mathit {b2}}+\mathit {a1} \mathit {b1} -2 \mathit {a2}}{2 \sqrt {\mathit {b1}^{2}-4 \mathit {b2}}}, \mathit {a1} , \sqrt {\mathit {b1}^{2}-4 \mathit {b2}}\, x \right )+\textit {\_C2} \,{\mathrm e}^{-\frac {x \left (\mathit {b1} +\sqrt {\mathit {b1}^{2}-4 \mathit {b2}}\right )}{2}} \KummerU \left (\frac {\mathit {a1} \sqrt {\mathit {b1}^{2}-4 \mathit {b2}}+\mathit {a1} \mathit {b1} -2 \mathit {a2}}{2 \sqrt {\mathit {b1}^{2}-4 \mathit {b2}}}, \mathit {a1} , \sqrt {\mathit {b1}^{2}-4 \mathit {b2}}\, x \right )\right ]\] 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))
Maple raw output
[y(x) = _C1*exp(-1/2*x*(b1+(b1^2-4*b2)^(1/2)))*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)+_C2*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)]