ODE
\[ -\left (a^2+1\right ) y(x)+y''(x)-2 \tan (x) y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0758277 (sec), leaf count = 48
\[\left \{\left \{y(x)\to \sec (x) \left (c_1 e^{-\sqrt {a^2} x}+\frac {c_2 e^{\sqrt {a^2} x}}{2 \sqrt {a^2}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.06 (sec), leaf count = 22
\[ \left \{ y \left ( x \right ) ={\frac {{\it \_C2}\,\cosh \left ( ax \right ) +{\it \_C1}\,\sinh \left ( ax \right ) }{\cos \left ( x \right ) }} \right \} \] Mathematica raw input
DSolve[-((1 + a^2)*y[x]) - 2*Tan[x]*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[1]/E^(Sqrt[a^2]*x) + (E^(Sqrt[a^2]*x)*C[2])/(2*Sqrt[a^2]))*Sec[x]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)-2*diff(y(x),x)*tan(x)-(a^2+1)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (_C2*cosh(a*x)+_C1*sinh(a*x))/cos(x)