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 = 0.22669 (sec), leaf count = 57

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

Maple ✓
cpu = 0.086 (sec), leaf count = 41

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

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

Mathematica raw output

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

Maple raw output

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