abreviatsjw
2021-12-28
Answered

Find the solution of the following differential equations.

$dy={e}^{x-y}dx$

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Esther Phillips

Answered 2021-12-29
Author has **34** answers

Step 1

$dy={e}^{x-y}dx$

$dy={e}^{x}\cdot {e}^{-y}dx$

$\frac{dy}{{e}^{-y}}={e}^{x}dz$

Step 2

${e}^{y}dy={e}^{x}dx$

Integrting both side

$\int {e}^{y}dy=\int {e}^{x}dx$ .

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

Step 2

Integrting both side

Jimmy Macias

Answered 2021-12-30
Author has **30** answers

In this tutorial we shall evaluate the simple differential equation of the form $\frac{dy}{dx}={e}^{x-y}$ using the method of separating the variables.

The differential equation of the form is given as

$\frac{dy}{dx}={e}^{x-y}$

$\Rightarrow \frac{dy}{dx}={e}^{x}{e}^{-y}$

$\Rightarrow \frac{dy}{dx}=\frac{{e}^{x}}{{e}^{y}}$

Separating the variables, the given differential equation can be written as

${e}^{y}dy={e}^{x}dx$ (i)

In the separating the variables technique we must keep the terms dy and dx in the numerators with their respective functions.

Now integrating both sides of the equation (i), we have

$\int {e}^{y}dy=\int {e}^{x}dx$

Using the formulas of integration$\int {e}^{x}dx={e}^{x}$ , we get

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

$\Rightarrow y=\mathrm{ln}({e}^{x}+c)$

This is the required solution of the given differential equation.

The differential equation of the form is given as

Separating the variables, the given differential equation can be written as

In the separating the variables technique we must keep the terms dy and dx in the numerators with their respective functions.

Now integrating both sides of the equation (i), we have

Using the formulas of integration

This is the required solution of the given differential equation.

Vasquez

Answered 2022-01-09
Author has **460** answers

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