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Question

(B) The given differential equation is $\frac{dy}{dx} + 2y = e^{-x}$.This is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$,where

Vedclass · Accepted Answer

$y e^{2x} = e^{x} + c$

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(B) The given differential equation is $\frac{dy}{dx} + 2y = e^{-x}$.This is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$,where $P = 2$ and $Q = e^{-x}$.The Integrating Factor ($I$.$F$.) is given by $I.F. = e^{\int P dx} = e^{\int 2 dx} = e^{2x}$.The general solution is given by $y \cdot (I.F.) = \int (Q \cdot I.F.) dx + c$.Substituting the values,we get $y e^{2x} = \int (e^{-x} \cdot e^{2x}) dx + c$.$y e^{2x} = \int e^{x} dx + c$.$y e^{2x} = e^{x} + c$.

The solution of the differential equation $\frac{dy}{dx} + 2y = e^{-x}$ is

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