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 = 0.013713 (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 = 0.143 (sec), leaf count = 45
\[ \left \{ y \left ( x \right ) ={1 \left ( {\it \_C2}\,{{\sl W}_{{\frac {-{\frac {i}{4}}a}{b}},\,{\frac {1}{4}}}\left (ib{x}^{2}\right )}+{\it \_C1}\,{{\sl M}_{{\frac {-{\frac {i}{4}}a}{b}},\,{\frac {1}{4}}}\left (ib{x}^{2}\right )} \right ) {\frac {1}{\sqrt {x}}}} \right \} \] 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[-(I*a + b)/(2*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),'implicit')
Maple raw output
y(x) = (_C2*WhittakerW(-1/4*I*a/b,1/4,I*b*x^2)+_C1*WhittakerM(-1/4*I*a/b,1/4,I*b*x^2))/x^(1/2)