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 = 0.00515818 (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 = 0.128 (sec), leaf count = 29

\[ \left \{ y \left ( x \right ) ={1 \left ( {\it \_C2}\,{{\sl W}_{-{\frac {a}{4}},\,{\frac {1}{4}}}\left ({x}^{2}\right )}+{\it \_C1}\,{{\sl M}_{-{\frac {a}{4}},\,{\frac {1}{4}}}\left ({x}^{2}\right )} \right ) {\frac {1}{\sqrt {x}}}} \right \} \] 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),'implicit')

Maple raw output

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