4.31.14 \(a y(x)+x^2 y''(x)+2 x^2 \cot (x) y'(x)=0\)

ODE
\[ a y(x)+x^2 y''(x)+2 x^2 \cot (x) y'(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0671049 (sec), leaf count = 50

\[\left \{\left \{y(x)\to \sqrt {x} \csc (x) \left (c_1 J_{\frac {1}{2} \sqrt {1-4 a}}(x)+c_2 Y_{\frac {1}{2} \sqrt {1-4 a}}(x)\right )\right \}\right \}\]

Maple ✓
cpu = 0.032 (sec), leaf count = 39

\[ \left \{ y \left ( x \right ) ={\frac {1}{\sin \left ( x \right ) }\sqrt {x} \left ( {{\sl J}_{{\frac {1}{2}\sqrt {1-4\,a}}}\left (x\right )}{\it \_C1}+{{\sl Y}_{{\frac {1}{2}\sqrt {1-4\,a}}}\left (x\right )}{\it \_C2} \right ) } \right \} \] Mathematica raw input

DSolve[a*y[x] + 2*x^2*Cot[x]*y'[x] + x^2*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> Sqrt[x]*(BesselJ[Sqrt[1 - 4*a]/2, x]*C[1] + BesselY[Sqrt[1 - 4*a]/2, x
]*C[2])*Csc[x]}}

Maple raw input

dsolve(x^2*diff(diff(y(x),x),x)+2*x^2*diff(y(x),x)*cot(x)+a*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = x^(1/2)*(BesselJ(1/2*(1-4*a)^(1/2),x)*_C1+BesselY(1/2*(1-4*a)^(1/2),x)*_C
2)/sin(x)