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Given the differential equation \\ y" + 25y = 7xe^{3x}, \ \ y(0) = 0, \ \ y'(0) = 0 \\ a) First,...

Question:

Given the differential equation

{eq}y'' + 25y = 7xe^{3x}, \ \ y(0) = 0, \ \ y'(0) = 0{/eq}

a) First, determine {eq}y_p{/eq}, the particular solution to the differential equation.

b) Next, determine {eq}y=y_c+y_p{/eq}, the complete solution to the differential equation.

c) Finally, solve the given initial value problem.

Inhomogeneous Linear Differential Equations:

Suppose that {eq}p_0(x),p_1(x),\dots,p_n(x), q(x) {/eq} are all functions, and neither {eq}p_n(x) {/eq} nor {eq}q(x) {/eq} is the zero function. Then the equation

{eq}\displaystyle \sum_{k=0}^n p_k(x)\frac{d^ky}{dx^k}=q(x) {/eq}

is called an {eq}n {/eq}th-order linear inhomogeneous equation for the unknown function {eq}y {/eq}. The function {eq}q(x) {/eq} is called the inhomogeneous term of this equation, and the equation

{eq}\displaystyle \sum_{k=0}^n p_k(x)\frac{d^ky}{dx^k}=0 {/eq}

is called the associated homogeneous equation.

Suppose that {eq}y_p {/eq} is any solution to a linear inhomogeneous differential equation. Then every solution to that inhomogeneous equation can be written in the form {eq}y=y_c+y_p {/eq} where {eq}y_c {/eq} is some solution to the associated homogeneous differential equation.

Answer and Explanation:

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(a) We'll use the method of undetermined coefficients to find a particular solution to the differential equation. The inhomogeneous term of the...

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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.


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