https://file.aiquickdraw.com/mathbot/file/17307237949974dgtzom4.jpg

To find a particular solution \( y_p \) to the differential equation:

\[
y'' + 16y = -16 \sin(4t),
\]

we will use the method of undetermined coefficients. Here’s the step-by-step process:

### Step 1: Identify the Form of the Particular Solution
Since the right side of the equation is \( -16 \sin(4t) \), we assume a particular solution of the form:

\[
y_p = A \sin(4t) + B \cos(4t),
\]

where \( A \) and \( B \) are constants to be determined.

### Step 2: Differentiate \( y_p \)
1. First derivative:
   \[
   y_p' = 4A \cos(4t) - 4B \sin(4t).
   \]

2. Second derivative:
   \[
   y_p'' = -16A \sin(4t) - 16B \cos(4t).
   \]

### Step 3: Substitute into the Differential Equation
Substitute \( y_p \) and \( y_p'' \) into the left side of the differential equation \( y'' + 16y \):

\[
y_p'' + 16y_p = (-16A \sin(4t) - 16B \cos(4t)) + 16(A \sin(4t) + B \cos(4t)).
\]

Simplifying, we get:

\[
y_p'' + 16y_p = 0.
\]

Since there is no way for the particular solution

Find a particular solution to y'' + 16y = -16 sin(4t).

This page provides a step-by-step solution for finding a particular solution to the differential equation y'' + 16y = -16 sin(4t) using the method of undetermined coefficients. Topics covered include differential equations and trigonometric functions, suitable for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation