Solution of the differential equation $\frac{d y}{d x}+2 y=e^{-x}$ is

Solution of the differential equation $\frac{d y}{d x}+2 y=e^{-x}$ is
  1. $y e^{x}=x+c$
  2. $y e^{2 x}=x+c$
  3. $y e^{x}=e^{2 x}+c$
  4. $y e^{2 x}=e^{x}+c$

Solution

$\frac{\mathrm{dy}}{\mathrm{dx}}+2 \mathrm{y}=\mathrm{e}^{-x}$ I.F. $=\mathrm{e}^{2 \int \mathrm{dx}}=\mathrm{e}^{2 \mathrm{x}}$ $\therefore \mathrm{ye}^{2 \mathrm{x}}=\int \mathrm{e}^{2 \mathrm{x}} \cdot \mathrm{e}^{-\mathrm{x}} \mathrm{dx}+\mathrm{c}$ $\therefore \mathrm{y} \mathrm{e}^{2 \mathrm{x}}=\int \mathrm{e}^{\mathrm{x}} \mathrm{dx}+\mathrm{c} \Rightarrow \mathrm{ye}^{2 \mathrm{x}}=\mathrm{e}^{\mathrm{x}}+\mathrm{c}$

Asked in: MHT CET 2020 (14 Oct Shift 2)

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