ODE
\[ 4 y''(x)=\left (a+x^2\right ) y(x) \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0200702 (sec), leaf count = 36
\[\left \{\left \{y(x)\to c_1 D_{\frac {1}{4} (-a-2)}(x)+c_2 D_{\frac {a-2}{4}}(i x)\right \}\right \}\]
Maple ✓
cpu = 0.148 (sec), leaf count = 33
\[ \left \{ y \left ( x \right ) ={1 \left ( {\it \_C2}\,{{\sl W}_{-{\frac {a}{8}},\,{\frac {1}{4}}}\left ({\frac {{x}^{2}}{2}}\right )}+{\it \_C1}\,{{\sl M}_{-{\frac {a}{8}},\,{\frac {1}{4}}}\left ({\frac {{x}^{2}}{2}}\right )} \right ) {\frac {1}{\sqrt {x}}}} \right \} \] Mathematica raw input
DSolve[4*y''[x] == (a + x^2)*y[x],y[x],x]
Mathematica raw output
{{y[x] -> C[1]*ParabolicCylinderD[(-2 - a)/4, x] + C[2]*ParabolicCylinderD[(-2 + a)/4, I*x]}}
Maple raw input
dsolve(4*diff(diff(y(x),x),x) = (x^2+a)*y(x), y(x),'implicit')
Maple raw output
y(x) = (_C2*WhittakerW(-1/8*a,1/4,1/2*x^2)+_C1*WhittakerM(-1/8*a,1/4,1/2*x^2))/x^(1/2)