ODE
\[ a y(x) \cos ^2(x)+y''(x)+\tan (x) y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0403675 (sec), leaf count = 30
\[\left \{\left \{y(x)\to c_2 \sin \left (\sqrt {a} \sin (x)\right )+c_1 \cos \left (\sqrt {a} \sin (x)\right )\right \}\right \}\]
Maple ✓
cpu = 0.119 (sec), leaf count = 23
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,\sin \left ( \sqrt {a}\sin \left ( x \right ) \right ) +{\it \_C2}\,\cos \left ( \sqrt {a}\sin \left ( x \right ) \right ) \right \} \] Mathematica raw input
DSolve[a*Cos[x]^2*y[x] + Tan[x]*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Cos[Sqrt[a]*Sin[x]] + C[2]*Sin[Sqrt[a]*Sin[x]]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+diff(y(x),x)*tan(x)+a*y(x)*cos(x)^2 = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*sin(a^(1/2)*sin(x))+_C2*cos(a^(1/2)*sin(x))