# Find: Solve the following separate differential equation ,Express the solution explicitly as a...

## Question:

Solve the following separate differential equation ,Express the solution explicitly as a function of x. {eq}\displaystyle \frac{dy}{dx}=e^{x-y} ,y(0)=\ln 3 {/eq}

## Differential Equations

The given equation is a first order differential equation. The most general form of a first-order differential equations is:

{eq}\frac{dy}{dx} = f(x,y) {/eq}.

If the differential equation is of the form {eq}P(y) \frac{dy}{dx} = Q(x) \\ P(y) dy = Q(x) dx {/eq} then it is a separable differential equation.

The particular solution can be obtained by integration on both sides.

Here are the useful formulas needed to solve the problem:

{eq}\int e^x dx = e^x + c\\ e^{\ln a} = a {/eq}

where c is the constant of integration.

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We are given the following equation along with the initial condition:

{eq}\displaystyle \frac{dy}{dx} = e^{x-y}\\ y(0) = \ln 3 {/eq}

First, we... First-Order Linear Differential Equations

from

Chapter 16 / Lesson 3
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In this lesson you'll learn how to solve a first-order linear differential equation. We first define what such an equation is, and then we give the algorithm for solving one of that form. Specific examples follow the more general description of the method.