##### 4.25.12 $$y(x) \left (a+b^2 x^2\right )+y''(x)=0$$

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

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 0.156673 (sec), leaf count = 62

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

Maple
cpu = 0.665 (sec), leaf count = 47

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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