What is a particular solution to the differential equation $\frac{dy}{dx}={e}^{x-y}$ with y(0)=2?

Sonia Rowland
2022-09-29
Answered

What is a particular solution to the differential equation $\frac{dy}{dx}={e}^{x-y}$ with y(0)=2?

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tiepidolu

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

this is separable

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

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

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

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

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

$y\left(0\right)=2\Rightarrow {e}^{2}=1+C\Rightarrow C={e}^{2}-1$

${e}^{y}={e}^{x}+{e}^{2}-1$

$y=\mathrm{ln}({e}^{x}+{e}^{2}-1)$

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

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

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

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

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

$y\left(0\right)=2\Rightarrow {e}^{2}=1+C\Rightarrow C={e}^{2}-1$

${e}^{y}={e}^{x}+{e}^{2}-1$

$y=\mathrm{ln}({e}^{x}+{e}^{2}-1)$

asked 2022-02-16

Suppose we have

$\frac{dy}{dx}+f\left(x\right)y=r\left(x\right)$

and it has two solutions${y}_{1}\left(x\right)$ and ${y}_{2}\left(x\right)$ then how to prove that solution of differential equation

$\frac{dy}{dx}+f\left(x\right)y=2r\left(x\right)$

Will be${y}_{1}\left(x\right)+{y}_{2}\left(x\right)$ ? I think given differential equations is linear first order equation so its solution will be

$y.{e}^{\int f\left(x\right)dx}=\int r.{e}^{\int f\left(x\right)dx}dx$

now do I establish two solution as$y}_{1$ and $y}_{2$ out of this equation?

and it has two solutions

Will be

now do I establish two solution as

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I have a simple, two object thermodynamic model with radiation and advection. This model consists of two first order quadratic differential equations, what I would like to solve analytically. The equations can be simplified to

$\frac{d}{dt}x={a}_{0}{x}^{4}+{a}_{1}{y}^{4}+{a}_{2}x+{a}_{3}y+{a}_{4}$

$\frac{d}{dt}y={a}_{5}{x}^{4}+{a}_{6}{y}^{4}+{a}_{7}x+{a}_{8}y+{a}_{9}$

I'm trying to find the analytic solution because I would like to use the model in an embedded system with limited resources, where solving it numerically is not feasible. So far I have tried to solve this with Maxima without any success. I have also consulted with my college textbooks, but they cover only linear differential equation systems.

$\frac{d}{dt}x={a}_{0}{x}^{4}+{a}_{1}{y}^{4}+{a}_{2}x+{a}_{3}y+{a}_{4}$

$\frac{d}{dt}y={a}_{5}{x}^{4}+{a}_{6}{y}^{4}+{a}_{7}x+{a}_{8}y+{a}_{9}$

I'm trying to find the analytic solution because I would like to use the model in an embedded system with limited resources, where solving it numerically is not feasible. So far I have tried to solve this with Maxima without any success. I have also consulted with my college textbooks, but they cover only linear differential equation systems.

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I have

$\dot{x}=\left(\begin{array}{c}{x}_{1}^{2}\\ -1\end{array}\right)$

I have

$\dot{x}=\left(\begin{array}{c}{x}_{1}^{2}\\ -1\end{array}\right)$

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