Solve the differential equation d y d x +y= e ^ - x - Vidyadip

Question

Solve the differential equation $\frac{ d y}{ d x}+y= e ^{- x }$

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Answer

$\frac{ d y}{ d x}+y= e ^{- x }$
The given equation is of the form
$\frac{ d y}{ d x}+ P y= Q \text {. }$
where $P=1$ and $Q=e^{-x}$
$ \therefore \text { I.F. }= e ^{\int^{ Pd x}}$
$= e ^{\int d x}$
$= e ^{ x } $
$\therefore$ Solution of the given equation is
$ y(\text { I.F. })=\int Q (\text { I.F. }) d x+ c$
$\therefore y \cdot e ^x=\int e ^{-x} \times e ^x d x+ c$
$\therefore y \cdot e ^x=\int e ^{-x+x} d x+ c$
$\therefore y \cdot e ^x=\int e ^0 d x+ c$
$\therefore y \cdot e ^x=\int 1 d x+ c$
$\therefore y e ^{ x }= x + c $

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