4.1.6 $y^{\prime }(x)=a\sin (bx+c)+ky(x)$

ODE
$$y^{\prime }(x)=a\sin (bx+c)+ky(x)$$ ODE Classification

[[_linear, `class A`]]

Book solution method
Linear ODE

Mathematica ✓
cpu = 0.0671456 (sec), leaf count = 43

$$\left\{\{y(x)\to c_1{e}^{kx}-\frac{a(k\sin (bx+c)+b\cos (bx+c))}{b^2+k^2}\}\right\}$$

Maple ✓
cpu = 0.013 (sec), leaf count = 40

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

DSolve[y'[x] == a*Sin[c + b*x] + k*y[x],y[x],x]

Mathematica raw output

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

Maple raw input

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

Maple raw output

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