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 = 0.162503 (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 = 0.615 (sec), leaf count = 35
\[\left [y \left (x \right ) = \frac {\textit {\_C1} \WhittakerM \left (-\frac {a}{8}, \frac {1}{4}, \frac {x^{2}}{2}\right )}{\sqrt {x}}+\frac {\textit {\_C2} \WhittakerW \left (-\frac {a}{8}, \frac {1}{4}, \frac {x^{2}}{2}\right )}{\sqrt {x}}\right ]\] 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))
Maple raw output
[y(x) = _C1/x^(1/2)*WhittakerM(-1/8*a,1/4,1/2*x^2)+_C2/x^(1/2)*WhittakerW(-1/8*a,1/4,1/2*x^2)]