y" + y' + 16y = 0. (a) Find the general solution of this differential equation. (b) Find the particular solution that satisfies the initial conditions y(0) = 3, y'(0) = 9. (c) Draw the graph of the particular solution for 0 < t < 15.
y" + y' + 16y = 0. (a) Find the general solution of this differential equation. (b) Find the particular solution that satisfies the initial conditions y(0) = 3, y'(0) = 9. (c) Draw the graph of the particular solution for 0 < t < 15.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Differential Equations
#### Problem Statement
Consider the second-order linear homogeneous differential equation given by:
\[ y'' + y' + 16y = 0. \]
(a) **Find the general solution of this differential equation.**
(b) **Find the particular solution that satisfies the initial conditions \( y(0) = 3 \) and \( y'(0) = 9 \).**
(c) **Draw the graph of the particular solution for \( 0 < t < 15 \).**
#### Explanation:
**(a) General Solution:**
To find the general solution of the differential equation \( y'' + y' + 16y = 0 \), we start by finding the characteristic equation. The characteristic equation is obtained by replacing \( y'' \) with \( r^2 \), \( y' \) with \( r \), and \( y \) with 1.
\[ r^2 + r + 16 = 0 \]
Solving this quadratic equation gives the roots of the characteristic equation.
**(b) Particular Solution:**
Next, we use the initial conditions \( y(0) = 3 \) and \( y'(0) = 9 \) to find the specific constants in our general solution from part (a).
**(c) Graph of the Particular Solution:**
Finally, to fully understand the solution, we can plot the particular solution for the range \( 0 < t < 15 \).
---
Graph: ***Explanation Required***
For part (c), you would draw the graph of the particular solution. The graph would typically illustrate how the solution behaves over time within the given interval \(0 < t < 15\).
---
This transcription provides a structured approach to solving the given differential equation, outlining the steps necessary for deriving both the general and particular solutions and understanding their graphical representations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd30ea730-ba6b-4d49-90df-80ffc6e77ea4%2F039b58d0-924f-453e-9c6d-cb6022d4800a%2F445rp9_processed.png&w=3840&q=75)
Transcribed Image Text:### Differential Equations
#### Problem Statement
Consider the second-order linear homogeneous differential equation given by:
\[ y'' + y' + 16y = 0. \]
(a) **Find the general solution of this differential equation.**
(b) **Find the particular solution that satisfies the initial conditions \( y(0) = 3 \) and \( y'(0) = 9 \).**
(c) **Draw the graph of the particular solution for \( 0 < t < 15 \).**
#### Explanation:
**(a) General Solution:**
To find the general solution of the differential equation \( y'' + y' + 16y = 0 \), we start by finding the characteristic equation. The characteristic equation is obtained by replacing \( y'' \) with \( r^2 \), \( y' \) with \( r \), and \( y \) with 1.
\[ r^2 + r + 16 = 0 \]
Solving this quadratic equation gives the roots of the characteristic equation.
**(b) Particular Solution:**
Next, we use the initial conditions \( y(0) = 3 \) and \( y'(0) = 9 \) to find the specific constants in our general solution from part (a).
**(c) Graph of the Particular Solution:**
Finally, to fully understand the solution, we can plot the particular solution for the range \( 0 < t < 15 \).
---
Graph: ***Explanation Required***
For part (c), you would draw the graph of the particular solution. The graph would typically illustrate how the solution behaves over time within the given interval \(0 < t < 15\).
---
This transcription provides a structured approach to solving the given differential equation, outlining the steps necessary for deriving both the general and particular solutions and understanding their graphical representations.
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