ODE
\[ (\text {a0}+\text {b0} x) y''(x)+(\text {a1}+\text {b1} x) y'(x)+y(x) (\text {a2}+\text {b2} x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.344159 (sec), leaf count = 307
\[\left \{\left \{y(x)\to e^{-\frac {x \left (\sqrt {\text {b1}^2-4 \text {b0} \text {b2}}+\text {b1}\right )}{2 \text {b0}}} (\text {a0}+\text {b0} x)^{\frac {\text {a0} \text {b1}-\text {a1} \text {b0}+\text {b0}^2}{\text {b0}^2}} \left (c_1 U\left (\frac {-2 \text {a2} \text {b0}^2+2 \sqrt {\text {b1}^2-4 \text {b0} \text {b2}} \text {b0}^2+2 \text {a0} \text {b2} \text {b0}+\text {a1} \left (\text {b1}-\sqrt {\text {b1}^2-4 \text {b0} \text {b2}}\right ) \text {b0}-\text {a0} \text {b1}^2+\text {a0} \text {b1} \sqrt {\text {b1}^2-4 \text {b0} \text {b2}}}{2 \text {b0}^2 \sqrt {\text {b1}^2-4 \text {b0} \text {b2}}},-\frac {\text {a1}}{\text {b0}}+\frac {\text {a0} \text {b1}}{\text {b0}^2}+2,\frac {\sqrt {\text {b1}^2-4 \text {b0} \text {b2}} (\text {a0}+\text {b0} x)}{\text {b0}^2}\right )+c_2 L_{\frac {2 \text {a2} \text {b0}^2-2 \sqrt {\text {b1}^2-4 \text {b0} \text {b2}} \text {b0}^2-2 \text {a0} \text {b2} \text {b0}+\text {a1} \left (\sqrt {\text {b1}^2-4 \text {b0} \text {b2}}-\text {b1}\right ) \text {b0}+\text {a0} \text {b1}^2-\text {a0} \text {b1} \sqrt {\text {b1}^2-4 \text {b0} \text {b2}}}{2 \text {b0}^2 \sqrt {\text {b1}^2-4 \text {b0} \text {b2}}}}^{\frac {\text {b0}^2-\text {a1} \text {b0}+\text {a0} \text {b1}}{\text {b0}^2}}\left (\frac {\sqrt {\text {b1}^2-4 \text {b0} \text {b2}} (\text {a0}+\text {b0} x)}{\text {b0}^2}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.559 (sec), leaf count = 287
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-\frac {\left (\sqrt {-4 \mathit {b2} \mathit {b0} +\mathit {b1}^{2}}+\mathit {b1} \right ) x}{2 \mathit {b0}}} \left (\mathit {b0} x +\mathit {a0} \right )^{\frac {\mathit {a0} \mathit {b1} -\mathit {a1} \mathit {b0} +\mathit {b0}^{2}}{\mathit {b0}^{2}}} \KummerM \left (\frac {\left (\mathit {a0} \mathit {b1} -\mathit {a1} \mathit {b0} +2 \mathit {b0}^{2}\right ) \sqrt {-4 \mathit {b2} \mathit {b0} +\mathit {b1}^{2}}-2 \mathit {a2} \,\mathit {b0}^{2}+\left (2 \mathit {a0} \mathit {b2} +\mathit {a1} \mathit {b1} \right ) \mathit {b0} -\mathit {a0} \,\mathit {b1}^{2}}{2 \sqrt {-4 \mathit {b2} \mathit {b0} +\mathit {b1}^{2}}\, \mathit {b0}^{2}}, \frac {\mathit {a0} \mathit {b1} -\mathit {a1} \mathit {b0} +2 \mathit {b0}^{2}}{\mathit {b0}^{2}}, \frac {\sqrt {-4 \mathit {b2} \mathit {b0} +\mathit {b1}^{2}}\, \left (\mathit {b0} x +\mathit {a0} \right )}{\mathit {b0}^{2}}\right )+\textit {\_C2} \,{\mathrm e}^{-\frac {\left (\sqrt {-4 \mathit {b2} \mathit {b0} +\mathit {b1}^{2}}+\mathit {b1} \right ) x}{2 \mathit {b0}}} \left (\mathit {b0} x +\mathit {a0} \right )^{\frac {\mathit {a0} \mathit {b1} -\mathit {a1} \mathit {b0} +\mathit {b0}^{2}}{\mathit {b0}^{2}}} \KummerU \left (\frac {\left (\mathit {a0} \mathit {b1} -\mathit {a1} \mathit {b0} +2 \mathit {b0}^{2}\right ) \sqrt {-4 \mathit {b2} \mathit {b0} +\mathit {b1}^{2}}-2 \mathit {a2} \,\mathit {b0}^{2}+\left (2 \mathit {a0} \mathit {b2} +\mathit {a1} \mathit {b1} \right ) \mathit {b0} -\mathit {a0} \,\mathit {b1}^{2}}{2 \sqrt {-4 \mathit {b2} \mathit {b0} +\mathit {b1}^{2}}\, \mathit {b0}^{2}}, \frac {\mathit {a0} \mathit {b1} -\mathit {a1} \mathit {b0} +2 \mathit {b0}^{2}}{\mathit {b0}^{2}}, \frac {\sqrt {-4 \mathit {b2} \mathit {b0} +\mathit {b1}^{2}}\, \left (\mathit {b0} x +\mathit {a0} \right )}{\mathit {b0}^{2}}\right )\right ]\] Mathematica raw input
DSolve[(a2 + b2*x)*y[x] + (a1 + b1*x)*y'[x] + (a0 + b0*x)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> ((a0 + b0*x)^((-(a1*b0) + b0^2 + a0*b1)/b0^2)*(C[1]*HypergeometricU[(-2*a2*b0^2 - a0*b1^2 + 2*a0*b0*b2 + 2*b0^2*Sqrt[b1^2 - 4*b0*b2] + a0*b1*Sqrt[b1^2 - 4*b0*b2] + a1*b0*(b1 - Sqrt[b1^2 - 4*b0*b2]))/(2*b0^2*Sqrt[b1^2 - 4*b0*b2]), 2 - a1/b0 + (a0*b1)/b0^2, (Sqrt[b1^2 - 4*b0*b2]*(a0 + b0*x))/b0^2] + C[2]*LaguerreL[(2*a2*b0^2 + a0*b1^2 - 2*a0*b0*b2 - 2*b0^2*Sqrt[b1^2 - 4*b0*b2] - a0*b1*Sqrt[b1^2 - 4*b0*b2] + a1*b0*(-b1 + Sqrt[b1^2 - 4*b0*b2]))/(2*b0^2*Sqrt[b1^2 - 4*b0*b2]), (-(a1*b0) + b0^2 + a0*b1)/b0^2, (Sqrt[b1^2 - 4*b0*b2]*(a0 + b0*x))/b0^2]))/E^(((b1 + Sqrt[b1^2 - 4*b0*b2])*x)/(2*b0))}}
Maple raw input
dsolve((b0*x+a0)*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/b0*((-4*b0*b2+b1^2)^(1/2)+b1)*x)*(b0*x+a0)^(1/b0^2*(a0*b1-a1*b0+b0^2))*KummerM(1/2*((a0*b1-a1*b0+2*b0^2)*(-4*b0*b2+b1^2)^(1/2)-2*a2*b0^2+(2*a0*b2+a1*b1)*b0-a0*b1^2)/(-4*b0*b2+b1^2)^(1/2)/b0^2,(a0*b1-a1*b0+2*b0^2)/b0^2,1/b0^2*(-4*b0*b2+b1^2)^(1/2)*(b0*x+a0))+_C2*exp(-1/2/b0*((-4*b0*b2+b1^2)^(1/2)+b1)*x)*(b0*x+a0)^(1/b0^2*(a0*b1-a1*b0+b0^2))*KummerU(1/2*((a0*b1-a1*b0+2*b0^2)*(-4*b0*b2+b1^2)^(1/2)-2*a2*b0^2+(2*a0*b2+a1*b1)*b0-a0*b1^2)/(-4*b0*b2+b1^2)^(1/2)/b0^2,(a0*b1-a1*b0+2*b0^2)/b0^2,1/b0^2*(-4*b0*b2+b1^2)^(1/2)*(b0*x+a0))]