4.3.40 \(y'(x)=\text {Csx}(x) y(x) \sec (x)+\sec ^2(x)\)

ODE
\[ y'(x)=\text {Csx}(x) y(x) \sec (x)+\sec ^2(x) \] ODE Classification

[_linear]

Book solution method
Linear ODE

Mathematica
cpu = 2.76097 (sec), leaf count = 54

\[\left \{\left \{y(x)\to e^{\int _1^x \text {Csx}(K[1]) \sec (K[1]) \, dK[1]} \left (\int _1^x \sec ^2(K[2]) e^{-\int _1^{K[2]} \text {Csx}(K[1]) \sec (K[1]) \, dK[1]} \, dK[2]+c_1\right )\right \}\right \}\]

Maple
cpu = 0.111 (sec), leaf count = 41

\[ \left \{ y \relax (x ) = \left (\int \!2\,{\frac {1}{\cos \left (2\,x \right ) +1}{{\rm e}^{-\int \!{\frac {{\it Csx} \relax (x ) }{\cos \relax (x ) }}\,{\rm d}x}}}\,{\rm d}x+{\it \_C1} \right ) {{\rm e}^{\int \!{\frac {{\it Csx} \relax (x ) }{\cos \relax (x ) }}\,{\rm d}x}} \right \} \] Mathematica raw input

DSolve[y'[x] == Sec[x]^2 + Csx[x]*Sec[x]*y[x],y[x],x]

Mathematica raw output

{{y[x] -> E^Integrate[Csx[K[1]]*Sec[K[1]], {K[1], 1, x}]*(C[1] + Integrate[Sec[K
[2]]^2/E^Integrate[Csx[K[1]]*Sec[K[1]], {K[1], 1, K[2]}], {K[2], 1, x}])}}

Maple raw input

dsolve(diff(y(x),x) = sec(x)^2+y(x)*sec(x)*Csx(x), y(x),'implicit')

Maple raw output

y(x) = (Int(2/(cos(2*x)+1)*exp(-Int(Csx(x)/cos(x),x)),x)+_C1)*exp(Int(Csx(x)/cos
(x),x))