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

ODE
$$y^{\prime }(x)=\text{Csx}(x)y(x)\sec (x)+{\sec }^2(x)$$ ODE Classification

[_linear]

Book solution method
Linear ODE

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

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

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

$$\{y\left(x\right)=(\int 2\frac{1}{\cos \left(2x\right)+1}{\mathrm{e}}^{-\int \frac{\mathit{C}\mathit{s}\mathit{x}\left(x\right)}{\cos \left(x\right)}\mathrm{d}x}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1}){\mathrm{e}}^{\int \frac{\mathit{C}\mathit{s}\mathit{x}\left(x\right)}{\cos \left(x\right)}\mathrm{d}x}\}$$ 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))