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 = 0.127287 (sec), leaf count = 307
\[\left \{\left \{y(x)\to e^{-\frac {x \left (\sqrt {\text {b1}^2-4 \text {b0} \text {b2}}+\text {b1}\right )}{2 \text {b0}}} (\text {a0}+\text {b0} x)^{\frac {\text {a0} \text {b1}-\text {a1} \text {b0}+\text {b0}^2}{\text {b0}^2}} \left (c_1 U\left (\frac {-2 \text {a2} \text {b0}^2+2 \sqrt {\text {b1}^2-4 \text {b0} \text {b2}} \text {b0}^2+2 \text {a0} \text {b2} \text {b0}+\text {a1} \left (\text {b1}-\sqrt {\text {b1}^2-4 \text {b0} \text {b2}}\right ) \text {b0}-\text {a0} \text {b1}^2+\text {a0} \text {b1} \sqrt {\text {b1}^2-4 \text {b0} \text {b2}}}{2 \text {b0}^2 \sqrt {\text {b1}^2-4 \text {b0} \text {b2}}},-\frac {\text {a1}}{\text {b0}}+\frac {\text {a0} \text {b1}}{\text {b0}^2}+2,\frac {\sqrt {\text {b1}^2-4 \text {b0} \text {b2}} (\text {a0}+\text {b0} x)}{\text {b0}^2}\right )+c_2 L_{\frac {-\text {a0} \text {b1} \sqrt {\text {b1}^2-4 \text {b0} \text {b2}}-2 \text {a0} \text {b0} \text {b2}+\text {a0} \text {b1}^2+\text {a1} \text {b0} \left (\sqrt {\text {b1}^2-4 \text {b0} \text {b2}}-\text {b1}\right )+2 \text {a2} \text {b0}^2-2 \text {b0}^2 \sqrt {\text {b1}^2-4 \text {b0} \text {b2}}}{2 \text {b0}^2 \sqrt {\text {b1}^2-4 \text {b0} \text {b2}}}}^{\frac {\text {a0} \text {b1}-\text {a1} \text {b0}+\text {b0}^2}{\text {b0}^2}}\left (\frac {(\text {a0}+\text {b0} x) \sqrt {\text {b1}^2-4 \text {b0} \text {b2}}}{\text {b0}^2}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.216 (sec), leaf count = 248
\[ \left \{ y \left ( x \right ) ={{\rm e}^{-{\frac {x}{2\,{\it b0}} \left ( \sqrt {-4\,{\it b2}\,{\it b0}+{{\it b1}}^{2}}+{\it b1} \right ) }}} \left ( {\it b0}\,x+{\it a0} \right ) ^{{\frac {{\it a0}\,{\it b1}-{\it a1}\,{\it b0}+{{\it b0}}^{2}}{{{\it b0}}^{2}}}} \left ( {{\sl U}\left ({\frac {1}{2\,{{\it b0}}^{2}} \left ( \left ( {\it a0}\,{\it b1}-{\it a1}\,{\it b0}+2\,{{\it b0}}^{2} \right ) \sqrt {-4\,{\it b2}\,{\it b0}+{{\it b1}}^{2}}-2\,{\it a2}\,{{\it b0}}^{2}+ \left ( 2\,{\it a0}\,{\it b2}+{\it a1}\,{\it b1} \right ) {\it b0}-{\it a0}\,{{\it b1}}^{2} \right ) {\frac {1}{\sqrt {-4\,{\it b2}\,{\it b0}+{{\it b1}}^{2}}}}},\,{\frac {{\it a0}\,{\it b1}-{\it a1}\,{\it b0}+2\,{{\it b0}}^{2}}{{{\it b0}}^{2}}},\,{\frac {{\it b0}\,x+{\it a0}}{{{\it b0}}^{2}}\sqrt {-4\,{\it b2}\,{\it b0}+{{\it b1}}^{2}}}\right )}{\it \_C2}+{{\sl M}\left ({\frac {1}{2\,{{\it b0}}^{2}} \left ( \left ( {\it a0}\,{\it b1}-{\it a1}\,{\it b0}+2\,{{\it b0}}^{2} \right ) \sqrt {-4\,{\it b2}\,{\it b0}+{{\it b1}}^{2}}-2\,{\it a2}\,{{\it b0}}^{2}+ \left ( 2\,{\it a0}\,{\it b2}+{\it a1}\,{\it b1} \right ) {\it b0}-{\it a0}\,{{\it b1}}^{2} \right ) {\frac {1}{\sqrt {-4\,{\it b2}\,{\it b0}+{{\it b1}}^{2}}}}},\,{\frac {{\it a0}\,{\it b1}-{\it a1}\,{\it b0}+2\,{{\it b0}}^{2}}{{{\it b0}}^{2}}},\,{\frac {{\it b0}\,x+{\it a0}}{{{\it b0}}^{2}}\sqrt {-4\,{\it b2}\,{\it b0}+{{\it b1}}^{2}}}\right )}{\it \_C1} \right ) \right \} \] 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),'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)