ODE
\[ -\left ((2 n+1)^2-4 x^2\right ) y(x)+4 x^2 y''(x)+4 x y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.393746 (sec), leaf count = 26
\[\left \{\left \{y(x)\to c_1 J_{n+\frac {1}{2}}(x)+c_2 Y_{n+\frac {1}{2}}(x)\right \}\right \}\]
Maple ✓
cpu = 0.038 (sec), leaf count = 19
\[\left [y \left (x \right ) = \textit {\_C1} \BesselJ \left (n +\frac {1}{2}, x\right )+\textit {\_C2} \BesselY \left (n +\frac {1}{2}, x\right )\right ]\] Mathematica raw input
DSolve[-(((1 + 2*n)^2 - 4*x^2)*y[x]) + 4*x*y'[x] + 4*x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> BesselJ[1/2 + n, x]*C[1] + BesselY[1/2 + n, x]*C[2]}}
Maple raw input
dsolve(4*x^2*diff(diff(y(x),x),x)+4*x*diff(y(x),x)-((1+2*n)^2-4*x^2)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*BesselJ(n+1/2,x)+_C2*BesselY(n+1/2,x)]