Solve the equation \(\frac{dy}{dx}-2y=x+\cos x\).
Solution: \begin{align*} && e^{-2x}y'-2e^{-2x}y &= e^{-2x}(x+\cos x) \\ \Rightarrow && e^{-2x}y &= \int e^{-2x}(x+\cos x) \d x \\ \Rightarrow && e^{-2x} y &= -e^{-2x} \frac{1}{20} \left (5 + 10 x + 8 \cos x - 4 \sin x \right) + C \\ \Rightarrow && y &= -\frac{1}{20} \left (5 + 10 x + 8 \cos x - 4 \sin x \right) +Ce^{2x} \end{align*}