Solve the differential equation $\displaystyle \frac{d \mathrm{y}}{d \mathrm{x}}=\cos \mathrm{x}-2 \mathrm{y}$.

The given differential equation can be written as: \[\frac{d \mathrm{y}}{d \mathrm{x}}+2 \mathrm{y}=\cos \mathrm{x}, \text { Taking } P=2, Q=\cos \mathrm{x} \] Integrating factor is given by, $\displaystyle I=\mathrm{e}^{\int 2 d \mathrm{x}}=\mathrm{e}^{2 \mathrm{x}}$ ∴ The solution is, $\displaystyle \mathrm{y} \cdot \mathrm{e}^{2 \mathrm{x}}=\int \mathrm{e}^{2 \mathrm{x}} \cos \mathrm{xdx}$ \[\text { Let, } \begin{aligned} I_{1} & =\int \cos \mathrm{x} \cdot \mathrm{e}^{2 \mathrm{x}} d \mathrm{x}=\cos \mathrm{x} \frac{\mathrm{e}^{2 \mathrm{x}}}{2}-\int(-\sin \mathrm{x}) \frac{\mathrm{e}^{2 \mathrm{x}}}{2} d \mathrm{x} \\ & =\frac{\mathrm{e}^{2 \mathrm{x}} \cos \mathrm{x}}{2}+\frac{1}{2}\left[\sin \mathrm{x} \cdot \frac{\mathrm{e}^{2 \mathrm{x}}}{2}-\int \cos \mathrm{x} \cdot \frac{\mathrm{e}^{2 \mathrm{x}}}{2} d \mathrm{x}\right] \\ \Rightarrow I_{1} & =\frac{\mathrm{e}^{2 \mathrm{x}} \cos \mathrm{x}}{2}+\frac{\mathrm{e}^{2 \mathrm{x}} \sin \mathrm{x}}{4}-\frac{1}{4} I_{1} \Rightarrow I_{1}=\frac{\mathrm{e}^{2 \mathrm{x}}}{5}(2 \cos \mathrm{x}+\sin \mathrm{x}) \end{aligned} \] ∴ The solution of the differential equation is \[\mathrm{y} \cdot \mathrm{e}^{2 \mathrm{x}}=\frac{\mathrm{e}^{2 \mathrm{x}}}{5}(2 \cos \mathrm{x}+\sin \mathrm{x})+C \Rightarrow \mathrm{y}=\frac{1}{5}(2 \cos \mathrm{x}+\sin \mathrm{x})+C \mathrm{e}^{-2 \mathrm{x}} \]