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
$\frac{d y}{d x}+p y=Q$
where, $P = 1$ and $Q = e^{-x}$
$\therefore$ I.F. $=e \int^{p d x}=e \int^{1 . d x}=e^x$
$\therefore $ Solution of the given equation is
$ y(I . F .)=\int Q(I . F .) d x+c$
$ \therefore y e^x=\int e^{-x} e^x d x+c$
$ \therefore y e^x=\int 1 d x+c$
$ \therefore ye ^{ x }= x + c $

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