Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Differential Equations: Solving Second-Order Initial Value Problems (IVPs)
#### Problem Statement:
Consider the second-order differential equation given by:
\[ x'' + x = 0. \]
A general solution to this differential equation is
\[ x(t) = c_1 \cos t + c_2 \sin t, \]
where \( c_1 \) and \( c_2 \) are constants. This general solution is a two-parameter family of solutions.
#### Given Initial Conditions:
- \( x(0) = -1 \)
- \( x'(0) = 7 \)
#### Task:
Find a particular solution of the differential equation that satisfies the given initial conditions.
#### Solution:
To solve this IVP, we need to determine the values of \( c_1 \) and \( c_2 \) that fit the initial conditions provided.
1. **Initial Condition:** \( x(0) = -1 \)
Substitute \( t = 0 \) into the general solution:
\[
x(0) = c_1 \cos(0) + c_2 \sin(0) = c_1 \cdot 1 + c_2 \cdot 0 = c_1
\]
Thus, we have \( c_1 = -1 \).
2. **Initial Condition:** \( x'(0) = 7 \)
First, find the derivative of the general solution:
\[
x'(t) = -c_1 \sin t + c_2 \cos t
\]
Now, substitute \( t = 0 \):
\[
x'(0) = -c_1 \sin(0) + c_2 \cos(0) = -c_1 \cdot 0 + c_2 \cdot 1 = c_2
\]
Thus, we have \( c_2 = 7 \).
Therefore, the particular solution that satisfies the initial conditions is:
\[ x(t) = -\cos t + 7 \sin t \]
#### User Interaction:
The website provides an input field for users to type in their answer and a button labeled "Read It" for additional help and explanations.
**Input Field:**
- \( x = \) [ ]
**Help Section (Button):**
- **Need Help?** [Read](https://content.bartleby.com/qna-images/question/0436118d-47b7-4fa9-abd3-dac72bbeccd1/a7ec6988-6b23-4185-b1f6-b895cc371ec8/pjy8bzq_processed.jpeg)
Transcribed Image Text:### Differential Equations: Solving Second-Order Initial Value Problems (IVPs)
#### Problem Statement:
Consider the second-order differential equation given by:
\[ x'' + x = 0. \]
A general solution to this differential equation is
\[ x(t) = c_1 \cos t + c_2 \sin t, \]
where \( c_1 \) and \( c_2 \) are constants. This general solution is a two-parameter family of solutions.
#### Given Initial Conditions:
- \( x(0) = -1 \)
- \( x'(0) = 7 \)
#### Task:
Find a particular solution of the differential equation that satisfies the given initial conditions.
#### Solution:
To solve this IVP, we need to determine the values of \( c_1 \) and \( c_2 \) that fit the initial conditions provided.
1. **Initial Condition:** \( x(0) = -1 \)
Substitute \( t = 0 \) into the general solution:
\[
x(0) = c_1 \cos(0) + c_2 \sin(0) = c_1 \cdot 1 + c_2 \cdot 0 = c_1
\]
Thus, we have \( c_1 = -1 \).
2. **Initial Condition:** \( x'(0) = 7 \)
First, find the derivative of the general solution:
\[
x'(t) = -c_1 \sin t + c_2 \cos t
\]
Now, substitute \( t = 0 \):
\[
x'(0) = -c_1 \sin(0) + c_2 \cos(0) = -c_1 \cdot 0 + c_2 \cdot 1 = c_2
\]
Thus, we have \( c_2 = 7 \).
Therefore, the particular solution that satisfies the initial conditions is:
\[ x(t) = -\cos t + 7 \sin t \]
#### User Interaction:
The website provides an input field for users to type in their answer and a button labeled "Read It" for additional help and explanations.
**Input Field:**
- \( x = \) [ ]
**Help Section (Button):**
- **Need Help?** [Read
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