##### 4.25.11 $$y''(x)=\left (a+x^2\right ) y(x)$$

ODE
$y''(x)=\left (a+x^2\right ) y(x)$ ODE Classiﬁcation

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 0.150689 (sec), leaf count = 47

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

Maple
cpu = 0.621 (sec), leaf count = 31

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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