# Solve the following initial-value problem: x' = x + e^{3 t}, x (0) = 2.

## Question:

Solve the following initial-value problem:

{eq}\displaystyle x' = x + e^{3 t},\ x (0) = 2 {/eq}.

## First-order Linear Differential Equation:

A differential equation of the form {eq}x' + p(t) x = q(t) {/eq} is a first-order linear differential equation. A formula for the solution is {eq}x(t) = e^{-\int p(t) \: dt} \left[ \int e^{\int p(t) \: dt} q(t) \: dt + C \right]. {/eq} To determine the value of {eq}C, {/eq} we need an initial condition.

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Subtracting {eq}x {/eq} from both sides gives the equation {eq}x' - x = e^{3t}, {/eq} so this is a linear first-order differential equation with...

First-Order Linear Differential Equations

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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.