ODE
\[ -\left (a^2 x^2+1\right ) y(x)+4 x^2 y''(x)+4 x y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.17349 (sec), leaf count = 35
\[\left \{\left \{y(x)\to \frac {e^{-\frac {a x}{2}} \left (c_2 e^{a x}+a c_1\right )}{a \sqrt {x}}\right \}\right \}\]
Maple ✓
cpu = 0.247 (sec), leaf count = 25
\[\left [y \left (x \right ) = \frac {\textit {\_C1} \sinh \left (\frac {a x}{2}\right )}{\sqrt {x}}+\frac {\textit {\_C2} \cosh \left (\frac {a x}{2}\right )}{\sqrt {x}}\right ]\] Mathematica raw input
DSolve[-((1 + a^2*x^2)*y[x]) + 4*x*y'[x] + 4*x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (a*C[1] + E^(a*x)*C[2])/(a*E^((a*x)/2)*Sqrt[x])}}
Maple raw input
dsolve(4*x^2*diff(diff(y(x),x),x)+4*x*diff(y(x),x)-(a^2*x^2+1)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1/x^(1/2)*sinh(1/2*a*x)+_C2/x^(1/2)*cosh(1/2*a*x)]