ODE
\[ a y(x) \cot ^2(x)+y''(x)+\tan (x) y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.305244 (sec), leaf count = 56
\[\left \{\left \{y(x)\to \left (-\sin ^2(x)\right )^{\frac {1}{4}-\frac {1}{4} \sqrt {1-4 a}} \left (c_2 \left (-\sin ^2(x)\right )^{\frac {1}{2} \sqrt {1-4 a}}+c_1\right )\right \}\right \}\]
Maple ✓
cpu = 0.171 (sec), leaf count = 37
\[ \left \{ y \left ( x \right ) ={\it \_C1}\, \left ( \sin \left ( x \right ) \right ) ^{{\frac {1}{2}\sqrt {1-4\,a}}+{\frac {1}{2}}}+{\it \_C2}\, \left ( \sin \left ( x \right ) \right ) ^{{\frac {1}{2}}-{\frac {1}{2}\sqrt {1-4\,a}}} \right \} \] Mathematica raw input
DSolve[a*Cot[x]^2*y[x] + Tan[x]*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-Sin[x]^2)^(1/4 - Sqrt[1 - 4*a]/4)*(C[1] + C[2]*(-Sin[x]^2)^(Sqrt[1 - 4*a]/2))}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+diff(y(x),x)*tan(x)+a*y(x)*cot(x)^2 = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*sin(x)^(1/2*(1-4*a)^(1/2)+1/2)+_C2*sin(x)^(1/2-1/2*(1-4*a)^(1/2))