How do you find the general solution to $\frac{dy}{dx}={e}^{x-y}$?

la1noxz
2022-09-29
Answered

How do you find the general solution to $\frac{dy}{dx}={e}^{x-y}$?

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Jeremy Mayo

Answered 2022-09-30
Author has **8** answers

$y\prime ={e}^{x-y}={e}^{x}{e}^{-y}$

so this is separable

$e}^{y}y\prime ={e}^{x$

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

$\int \frac{d}{dx}\left({e}^{y}\right)dx=\int {e}^{x}dx$

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

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

so this is separable

$e}^{y}y\prime ={e}^{x$

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

$\int \frac{d}{dx}\left({e}^{y}\right)dx=\int {e}^{x}dx$

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

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

asked 2022-06-20

For example, if I have the system,

${y}^{\prime}+y=3x\phantom{\rule{0ex}{0ex}}{y}^{\prime}-y=x$

I could then use elimination to minus the top equation from the bottom one to get,

$2y=2x\phantom{\rule{0ex}{0ex}}y=x$

Which is obviously wrong as then, $1+x=3x$ which is wrong.

So why are you not able to use elimination in solving a system of first order differential equations?

${y}^{\prime}+y=3x\phantom{\rule{0ex}{0ex}}{y}^{\prime}-y=x$

I could then use elimination to minus the top equation from the bottom one to get,

$2y=2x\phantom{\rule{0ex}{0ex}}y=x$

Which is obviously wrong as then, $1+x=3x$ which is wrong.

So why are you not able to use elimination in solving a system of first order differential equations?

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a. Second order nonlinear differential equation.

b. First order linear differential equation.

c. Second order linear differential equation.

d. First order nonlinear differential equation.

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What is the general solution of the differential equation? : $(x-4){y}^{4}dx-{x}^{3}({y}^{2}-3)dy=0$

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What is a solution to the differential equation $\frac{dy}{dx}=3y$?

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Make use of vectors to re-write the following second order differential equation into a 1st order differential equation. (note: I do not need to solve it).

$2.1{x}^{-2}\frac{{d}^{2}y}{{dx}^{2}}-3{e}^{2x}{y}^{4}\frac{dy}{dx}=-6y$

where$y=4.2$ and $\frac{dy}{dx}=-3.1$ when $x=0.5$ .

Im not sure were to begin any help is greatly appreciated.

where

Im not sure were to begin any help is greatly appreciated.

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$xy\prime -3y=x-1$ Solve