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 = 0.151992 (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 = 0.666 (sec), leaf count = 39

\[\left [y \left (x \right ) = \frac {\textit {\_C1} \WhittakerM \left (-\frac {i a}{4}, \frac {1}{4}, i x^{2}\right )}{\sqrt {x}}+\frac {\textit {\_C2} \WhittakerW \left (-\frac {i a}{4}, \frac {1}{4}, i x^{2}\right )}{\sqrt {x}}\right ]\] Mathematica raw input

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

Mathematica raw output

{{y[x] -> C[1]*ParabolicCylinderD[(-1/2*I)*(-I + a), (1 + I)*x] + C[2]*Parabolic
CylinderD[(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))

Maple raw output

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