4.29.5 \((a+x) y''(x)+(\text {a1}+\text {b1} x) y'(x)+y(x) (\text {a2}+\text {b2} x)=0\)

ODE
\[ (a+x) y''(x)+(\text {a1}+\text {b1} x) y'(x)+y(x) (\text {a2}+\text {b2} x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0388146 (sec), leaf count = 239

\[\left \{\left \{y(x)\to e^{-\frac {1}{2} x \left (\sqrt {\text {b1}^2-4 \text {b2}}+\text {b1}\right )} (a+x)^{a \text {b1}-\text {a1}+1} \left (c_1 U\left (\frac {-a \text {b1}^2+\text {a1} \text {b1}+a \sqrt {\text {b1}^2-4 \text {b2}} \text {b1}-2 \text {a2}+2 a \text {b2}-\text {a1} \sqrt {\text {b1}^2-4 \text {b2}}+2 \sqrt {\text {b1}^2-4 \text {b2}}}{2 \sqrt {\text {b1}^2-4 \text {b2}}},-\text {a1}+a \text {b1}+2,\sqrt {\text {b1}^2-4 \text {b2}} (a+x)\right )+c_2 L_{\frac {-a \text {b1} \sqrt {\text {b1}^2-4 \text {b2}}+a \text {b1}^2-2 a \text {b2}+\text {a1} \sqrt {\text {b1}^2-4 \text {b2}}-\text {a1} \text {b1}+2 \text {a2}-2 \sqrt {\text {b1}^2-4 \text {b2}}}{2 \sqrt {\text {b1}^2-4 \text {b2}}}}^{a \text {b1}-\text {a1}+1}\left ((a+x) \sqrt {\text {b1}^2-4 \text {b2}}\right )\right )\right \}\right \}\]

Maple ✓
cpu = 0.164 (sec), leaf count = 171

\[ \left \{ y \left ( x \right ) ={{\rm e}^{-{\frac {x}{2} \left ( {\it b1}+\sqrt {{{\it b1}}^{2}-4\,{\it b2}} \right ) }}} \left ( a+x \right ) ^{{\it b1}\,a-{\it a1}+1} \left ( {{\sl U}\left (-{\frac {1}{2} \left ( \left ( -{\it b1}\,a+{\it a1}-2 \right ) \sqrt {{{\it b1}}^{2}-4\,{\it b2}}+a{{\it b1}}^{2}-2\,a{\it b2}-{\it a1}\,{\it b1}+2\,{\it a2} \right ) {\frac {1}{\sqrt {{{\it b1}}^{2}-4\,{\it b2}}}}},\,{\it b1}\,a-{\it a1}+2,\,\sqrt {{{\it b1}}^{2}-4\,{\it b2}} \left ( a+x \right ) \right )}{\it \_C2}+{{\sl M}\left (-{\frac {1}{2} \left ( \left ( -{\it b1}\,a+{\it a1}-2 \right ) \sqrt {{{\it b1}}^{2}-4\,{\it b2}}+a{{\it b1}}^{2}-2\,a{\it b2}-{\it a1}\,{\it b1}+2\,{\it a2} \right ) {\frac {1}{\sqrt {{{\it b1}}^{2}-4\,{\it b2}}}}},\,{\it b1}\,a-{\it a1}+2,\,\sqrt {{{\it b1}}^{2}-4\,{\it b2}} \left ( a+x \right ) \right )}{\it \_C1} \right ) \right \} \] Mathematica raw input

DSolve[(a2 + b2*x)*y[x] + (a1 + b1*x)*y'[x] + (a + x)*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> ((a + x)^(1 - a1 + a*b1)*(C[1]*HypergeometricU[(-2*a2 + a1*b1 - a*b1^2
 + 2*Sqrt[b1^2 - 4*b2] - a1*Sqrt[b1^2 - 4*b2] + a*b1*Sqrt[b1^2 - 4*b2] + 2*a*b2)
/(2*Sqrt[b1^2 - 4*b2]), 2 - a1 + a*b1, Sqrt[b1^2 - 4*b2]*(a + x)] + C[2]*Laguerr
eL[(2*a2 - a1*b1 + a*b1^2 - 2*Sqrt[b1^2 - 4*b2] + a1*Sqrt[b1^2 - 4*b2] - a*b1*Sq
rt[b1^2 - 4*b2] - 2*a*b2)/(2*Sqrt[b1^2 - 4*b2]), 1 - a1 + a*b1, Sqrt[b1^2 - 4*b2
]*(a + x)]))/E^(((b1 + Sqrt[b1^2 - 4*b2])*x)/2)}}

Maple raw input

dsolve((a+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)))*(a+x)^(a*b1-a1+1)*(KummerU(-1/2/(b1^2-
4*b2)^(1/2)*((-a*b1+a1-2)*(b1^2-4*b2)^(1/2)+a*b1^2-2*a*b2-a1*b1+2*a2),a*b1-a1+2,
(b1^2-4*b2)^(1/2)*(a+x))*_C2+KummerM(-1/2/(b1^2-4*b2)^(1/2)*((-a*b1+a1-2)*(b1^2-
4*b2)^(1/2)+a*b1^2-2*a*b2-a1*b1+2*a2),a*b1-a1+2,(b1^2-4*b2)^(1/2)*(a+x))*_C1)