4.25.9 \(\left (a+x^2\right ) y(x)+y''(x)=0\)

ODE
\[ \left (a+x^2\right ) y(x)+y''(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.00677539 (sec), leaf count = 46

\[\left \{\left \{y(x)\to c_1 D_{-\frac {1}{2} i (a-i)}((1+i) x)+c_2 D_{\frac {1}{2} i (a+i)}((-1+i) x)\right \}\right \}\]

Maple ✓
cpu = 0.307 (sec), leaf count = 37

\[ \left \{ y \left ( x \right ) ={1 \left ( {\it \_C2}\,{{\sl W}_{-{\frac {i}{4}}a,\,{\frac {1}{4}}}\left (i{x}^{2}\right )}+{\it \_C1}\,{{\sl M}_{-{\frac {i}{4}}a,\,{\frac {1}{4}}}\left (i{x}^{2}\right )} \right ) {\frac {1}{\sqrt {x}}}} \right \} \] Mathematica raw input

DSolve[(a + x^2)*y[x] + y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> C[1]*ParabolicCylinderD[(-I/2)*(-I + a), (1 + I)*x] + C[2]*ParabolicCy
linderD[(I/2)*(I + a), (-1 + I)*x]}}

Maple raw input

dsolve(diff(diff(y(x),x),x)+(x^2+a)*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = (_C2*WhittakerW(-1/4*I*a,1/4,I*x^2)+_C1*WhittakerM(-1/4*I*a,1/4,I*x^2))/x
^(1/2)