4.27.34 \(y(x) \left (a \cos ^2(x)-\sec ^2(x)\right )+y''(x)-\tan (x) y'(x)=0\)

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 = 0.635292 (sec), leaf count = 86

\[\left \{\left \{y(x)\to \sec (x) e^{\sqrt {a} \sqrt {\cos (x)-1} \sqrt {\cos (x)+1}} \left (c_2 \int _1^{\cos (x)}\frac {e^{-2 \sqrt {a} \sqrt {K[1]-1} \sqrt {K[1]+1}} K[1]}{\sqrt {1-K[1]^2}}dK[1]+c_1\right )\right \}\right \}\]

Maple ✓
cpu = 0.731 (sec), leaf count = 35

\[\left [y \left (x \right ) = \frac {\textit {\_C1} \sinh \left (\sqrt {-a}\, \sin \left (x \right )\right )}{\cos \left (x \right )}+\frac {\textit {\_C2} \cosh \left (\sqrt {-a}\, \sin \left (x \right )\right )}{\cos \left (x \right )}\right ]\] 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

{{y[x] -> E^(Sqrt[a]*Sqrt[-1 + Cos[x]]*Sqrt[1 + Cos[x]])*Sec[x]*(C[1] + C[2]*Ina
ctive[Integrate][K[1]/(E^(2*Sqrt[a]*Sqrt[-1 + K[1]]*Sqrt[1 + K[1]])*Sqrt[1 - K[1
]^2]), {K[1], 1, Cos[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))

Maple raw output

[y(x) = _C1/cos(x)*sinh((-a)^(1/2)*sin(x))+_C2/cos(x)*cosh((-a)^(1/2)*sin(x))]