4.2.21 $y^{\prime }(x)=y(x)(a+by(x)\cos (kx))$

ODE
$$y^{\prime }(x)=y(x)(a+by(x)\cos (kx))$$ ODE Classification

[_Bernoulli]

Book solution method
The Bernoulli ODE

Mathematica ✓
cpu = 0.0492755 (sec), leaf count = 57

$$\left\{\{y(x)\to -\frac{(a^2+k^2)e^{ax}}{c_1(-(a^2+k^2))+bk{e}^{ax}\sin (kx)+ab{e}^{ax}\cos (kx)}\}\right\}$$

Maple ✓
cpu = 0.016 (sec), leaf count = 54

$$\{{\left(y\left(x\right)\right)}^{-1}+\frac{b{\mathrm{e}}^{ax}(\cos \left(kx\right)a+k\sin \left(kx\right))-\mathit{\_}\mathit{C}\mathit{1}(a^2+k^2)}{(a^2+k^2){\mathrm{e}}^{ax}}=0\}$$ Mathematica raw input

DSolve[y'[x] == y[x]*(a + b*Cos[k*x]*y[x]),y[x],x]

Mathematica raw output

{{y[x] -> -((E^(a*x)*(a^2 + k^2))/(-((a^2 + k^2)*C[1]) + a*b*E^(a*x)*Cos[k*x] + 
b*E^(a*x)*k*Sin[k*x]))}}

Maple raw input

dsolve(diff(y(x),x) = (a+b*y(x)*cos(k*x))*y(x), y(x),'implicit')

Maple raw output

1/y(x)+(b*exp(a*x)*(cos(k*x)*a+k*sin(k*x))-_C1*(a^2+k^2))/(a^2+k^2)/exp(a*x) = 0