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 = 0.0485332 (sec), leaf count = 57
\[\left \{\left \{y(x)\to \frac {e^{-\frac {1}{2} \sqrt {a^2} x} \left (c_2 e^{\sqrt {a^2} x}+\sqrt {a^2} c_1\right )}{\sqrt {a^2} \sqrt {x}}\right \}\right \}\]
Maple ✓
cpu = 0.117 (sec), leaf count = 23
\[ \left \{ y \left ( x \right ) ={1 \left ( {\it \_C1}\,\sinh \left ( {\frac {ax}{2}} \right ) +{\it \_C2}\,\cosh \left ( {\frac {ax}{2}} \right ) \right ) {\frac {1}{\sqrt {x}}}} \right \} \] 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] -> (Sqrt[a^2]*C[1] + E^(Sqrt[a^2]*x)*C[2])/(Sqrt[a^2]*E^((Sqrt[a^2]*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),'implicit')
Maple raw output
y(x) = (_C1*sinh(1/2*a*x)+_C2*cosh(1/2*a*x))/x^(1/2)