4.28.4 \((a+x) y(x)+x y''(x)=0\)

ODE
\[ (a+x) y(x)+x y''(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.12342 (sec), leaf count = 53

\[\left \{\left \{y(x)\to e^{-i x} x \left (c_2 \, _1F_1\left (\frac {i a}{2}+1;2;2 i x\right )+c_1 U\left (\frac {i a}{2}+1,2,2 i x\right )\right )\right \}\right \}\]

Maple ✓
cpu = 0.161 (sec), leaf count = 29

\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\sl M}_{-{\frac {i}{2}}a,\,{\frac {1}{2}}}\left (2\,ix\right )}+{\it \_C2}\,{{\sl W}_{-{\frac {i}{2}}a,\,{\frac {1}{2}}}\left (2\,ix\right )} \right \} \] Mathematica raw input

DSolve[(a + x)*y[x] + x*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (x*(C[2]*Hypergeometric1F1[1 + (I/2)*a, 2, (2*I)*x] + C[1]*Hypergeomet
ricU[1 + (I/2)*a, 2, (2*I)*x]))/E^(I*x)}}

Maple raw input

dsolve(x*diff(diff(y(x),x),x)+(a+x)*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = _C1*WhittakerM(-1/2*I*a,1/2,2*I*x)+_C2*WhittakerW(-1/2*I*a,1/2,2*I*x)