ODE
\[ y(x) \left (a \cos ^2(x)-\sec ^2(x)\right )+y''(x)-\tan (x) y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 200.591 (sec), leaf count = 0 , could not solve
DSolve[(a*Cos[x]^2 - Sec[x]^2)*y[x] - Tan[x]*Derivative[1][y][x] + Derivative[2][y][x] == 0, y[x], x]
Maple ✓
cpu = 0.327 (sec), leaf count = 32
\[ \left \{ y \left ( x \right ) ={\frac {1}{\cos \left ( x \right ) } \left ( {\it \_C1}\,\sinh \left ( \sqrt {-a}\sin \left ( x \right ) \right ) +{\it \_C2}\,\cosh \left ( \sqrt {-a}\sin \left ( x \right ) \right ) \right ) } \right \} \] Mathematica raw input
DSolve[(a*Cos[x]^2 - Sec[x]^2)*y[x] - Tan[x]*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[(a*Cos[x]^2 - Sec[x]^2)*y[x] - Tan[x]*Derivative[1][y][x] + Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve(diff(diff(y(x),x),x)-diff(y(x),x)*tan(x)+(a*cos(x)^2-sec(x)^2)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (_C1*sinh((-a)^(1/2)*sin(x))+_C2*cosh((-a)^(1/2)*sin(x)))/cos(x)