Key to "Solve differential equation dy+5ydx=e−5xdx"

Name: Solve differential equation dy+5ydx=e−5xdx
Uploaded: 2022-02-23T17:22:57+00:00
Description: Solve differential equation dy+5ydx=e−5xdx

College
Calculus and Analysis
Differential Equations
First order differential equations

OlmekinjP

Answered question

2021-01-13

Solve differential equation $d y + 5 y d x = e^{- 5 x} d x$

Answer & Explanation

d2saint0

Skilled2021-01-14Added 89 answers

$\frac{d y}{d x} + P (x) y = Q (x)$ (1)
where P(x) or Q(x) are constants or function of x alone
Integrating factor of (1) is
$I . F . = e^{\int P (x) d x}$
Required solution is
$y (I . F .) = \int Q (x) (I . F) d x + C$
$d y + (5 y) d x = e^{- 5 x} d x$
$\Rightarrow \frac{d y}{d x} + 5 y = e^{- 5 x} (*)$
Since this equation is in the form $\frac{d y}{d x} + P (x) y = Q (x)$ so it is linear ODE comparing $(*)$ with $\frac{d y}{d x} + P (x) y = Q (x)$ we get $P (x) = 5$ , $Q (x) = e^{- 5 x}$
So integrating factor is
$I . F . = e^{\int P (x) d x}$
$= e^{\int 5 d x}$
$= e^{5 x}$
$y (I . F .) = \int Q (x) (I . F .) d x + c$
$\Rightarrow y (e^{5 x}) = \int (e^{5 x}) (e^{5 x}) d x + C$
$\Rightarrow y (e^{5 x}) = \int d x + C$
$= y (e^{5 x}) = x + C$
$\Rightarrow y = \frac{x + C}{e^{5 x}}$

Jeffrey Jordon

Expert2021-12-25Added 2605 answers

Answer is given below (on video)

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