4.29.18 $(\text{a0}+\text{b0}x)y^{″}(x)+(\text{a1}+\text{b1}x)y^{\prime }(x)+y(x)(\text{a2}+\text{b2}x)=0$

ODE
$$(\text{a0}+\text{b0}x)y^{″}(x)+(\text{a1}+\text{b1}x)y^{\prime }(x)+y(x)(\text{a2}+\text{b2}x)=0$$ ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.127287 (sec), leaf count = 307

$$\left\{\{y(x)\to e^{-\frac{x(\sqrt{{\text{b1}}^2-4\text{b0}\text{b2}}+\text{b1})}{2\text{b0}}}(\text{a0}+\text{b0}x{)}^{\frac{\text{a0}\text{b1}-\text{a1}\text{b0}+{\text{b0}}^2}{{\text{b0}}^2}}(c_{1}U(\frac{-2\text{a2}{\text{b0}}^2+2\sqrt{{\text{b1}}^2-4\text{b0}\text{b2}}{\text{b0}}^2+2\text{a0}\text{b2}\text{b0}+\text{a1}(\text{b1}-\sqrt{{\text{b1}}^2-4\text{b0}\text{b2}})\text{b0}-\text{a0}{\text{b1}}^2+\text{a0}\text{b1}\sqrt{{\text{b1}}^2-4\text{b0}\text{b2}}}{2{\text{b0}}^2\sqrt{{\text{b1}}^2-4\text{b0}\text{b2}}},-\frac{\text{a1}}{\text{b0}}+\frac{\text{a0}\text{b1}}{{\text{b0}}^2}+2,\frac{\sqrt{{\text{b1}}^2-4\text{b0}\text{b2}}(\text{a0}+\text{b0}x)}{{\text{b0}}^2})+c_2{L}_{\frac{-\text{a0}\text{b1}\sqrt{{\text{b1}}^2-4\text{b0}\text{b2}}-2\text{a0}\text{b0}\text{b2}+\text{a0}{\text{b1}}^2+\text{a1}\text{b0}(\sqrt{{\text{b1}}^2-4\text{b0}\text{b2}}-\text{b1})+2\text{a2}{\text{b0}}^2-2{\text{b0}}^2\sqrt{{\text{b1}}^2-4\text{b0}\text{b2}}}{2{\text{b0}}^2\sqrt{{\text{b1}}^2-4\text{b0}\text{b2}}}}^{\frac{\text{a0}\text{b1}-\text{a1}\text{b0}+{\text{b0}}^2}{{\text{b0}}^2}}\left(\frac{(\text{a0}+\text{b0}x)\sqrt{{\text{b1}}^2-4\text{b0}\text{b2}}}{{\text{b0}}^2}\right))\}\right\}$$

Maple ✓
cpu = 0.216 (sec), leaf count = 248

$$\{y\left(x\right)={\mathrm{e}}^{-\frac{x}{2\mathit{b}\mathit{0}}(\sqrt{-4\mathit{b}\mathit{2}\mathit{b}\mathit{0}+{\mathit{b}\mathit{1}}^2}+\mathit{b}\mathit{1})}{(\mathit{b}\mathit{0}x+\mathit{a}\mathit{0})}^{\frac{\mathit{a}\mathit{0}\mathit{b}\mathit{1}-\mathit{a}\mathit{1}\mathit{b}\mathit{0}+{\mathit{b}\mathit{0}}^2}{{\mathit{b}\mathit{0}}^2}}(\mathit{U}(\frac{1}{2{\mathit{b}\mathit{0}}^2}((\mathit{a}\mathit{0}\mathit{b}\mathit{1}-\mathit{a}\mathit{1}\mathit{b}\mathit{0}+2{\mathit{b}\mathit{0}}^2)\sqrt{-4\mathit{b}\mathit{2}\mathit{b}\mathit{0}+{\mathit{b}\mathit{1}}^2}-2\mathit{a}\mathit{2}{\mathit{b}\mathit{0}}^2+(2\mathit{a}\mathit{0}\mathit{b}\mathit{2}+\mathit{a}\mathit{1}\mathit{b}\mathit{1})\mathit{b}\mathit{0}-\mathit{a}\mathit{0}{\mathit{b}\mathit{1}}^2)\frac{1}{\sqrt{-4\mathit{b}\mathit{2}\mathit{b}\mathit{0}+{\mathit{b}\mathit{1}}^2}},\frac{\mathit{a}\mathit{0}\mathit{b}\mathit{1}-\mathit{a}\mathit{1}\mathit{b}\mathit{0}+2{\mathit{b}\mathit{0}}^2}{{\mathit{b}\mathit{0}}^2},\frac{\mathit{b}\mathit{0}x+\mathit{a}\mathit{0}}{{\mathit{b}\mathit{0}}^2}\sqrt{-4\mathit{b}\mathit{2}\mathit{b}\mathit{0}+{\mathit{b}\mathit{1}}^2})\mathit{\_}\mathit{C}\mathit{2}+\mathit{M}(\frac{1}{2{\mathit{b}\mathit{0}}^2}((\mathit{a}\mathit{0}\mathit{b}\mathit{1}-\mathit{a}\mathit{1}\mathit{b}\mathit{0}+2{\mathit{b}\mathit{0}}^2)\sqrt{-4\mathit{b}\mathit{2}\mathit{b}\mathit{0}+{\mathit{b}\mathit{1}}^2}-2\mathit{a}\mathit{2}{\mathit{b}\mathit{0}}^2+(2\mathit{a}\mathit{0}\mathit{b}\mathit{2}+\mathit{a}\mathit{1}\mathit{b}\mathit{1})\mathit{b}\mathit{0}-\mathit{a}\mathit{0}{\mathit{b}\mathit{1}}^2)\frac{1}{\sqrt{-4\mathit{b}\mathit{2}\mathit{b}\mathit{0}+{\mathit{b}\mathit{1}}^2}},\frac{\mathit{a}\mathit{0}\mathit{b}\mathit{1}-\mathit{a}\mathit{1}\mathit{b}\mathit{0}+2{\mathit{b}\mathit{0}}^2}{{\mathit{b}\mathit{0}}^2},\frac{\mathit{b}\mathit{0}x+\mathit{a}\mathit{0}}{{\mathit{b}\mathit{0}}^2}\sqrt{-4\mathit{b}\mathit{2}\mathit{b}\mathit{0}+{\mathit{b}\mathit{1}}^2})\mathit{\_}\mathit{C}\mathit{1})\}$$ 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]*Laguer
reL[(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),'implicit')

Maple raw output

y(x) = 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))*_C2+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))*_C1)