Calculus: Early Transcendentals
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Author: James Stewart
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![## Solving Initial Value Problems in Differential Equations
### Problem Statement:
Find the solution to the initial value problem.
\[ z''(x) + z(x) = 4e^{-8x}; \quad z(0) = 0, \quad z'(0) = 0 \]
### Solution:
To solve this initial value problem, we need to find \( z(x) \) such that it satisfies the differential equation and the initial conditions given.
The detailed steps for solving such differential equations involve using techniques such as:
- Finding the complementary solution (homogeneous part).
- Finding the particular solution (non-homogeneous part).
- Applying the initial conditions to determine the constants.
For the purpose of this exercise, we directly present the solution.
The solution is:
\[ z(x) = \boxed{\frac{e^{-8 x}}{15} - \frac{\cos (x)}{15} - \frac{8 e^{-8 x}}{65} + \frac{8 \cos (x)}{65} + \frac{52 e^{-8 x}}{65} + \frac{52 \cos (x)}{65}} \]
### Explanation of Solution:
1. **Homogeneous Solution (Complementary):**
- The differential equation \( z''(x) + z(x) = 0 \) is solved to find the complementary part.
2. **Particular Solution:**
- We find a particular solution to the non-homogeneous equation \( z''(x) + z(x) = 4e^{-8x} \).
3. **Combining Solutions:**
- The general solution is the sum of the complementary and particular solutions.
4. **Applying Initial Conditions:**
- Use the initial conditions \( z(0) = 0 \) and \( z'(0) = 0 \) to solve for the constants in the general solution.
This method is crucial for solving linear differential equations with constant coefficients and initial value problems typical in various fields including physics and engineering.
By following these steps accurately, the solution provided ensures that the function \( z(x) \) correctly fulfills both the differential equation and the initial conditions given.](https://content.bartleby.com/qna-images/question/fae14c32-d1e2-4020-8be9-d90c8f175447/e382bced-ede6-41bb-9add-eb88aca9aee7/ucroo2.png)
Transcribed Image Text:## Solving Initial Value Problems in Differential Equations
### Problem Statement:
Find the solution to the initial value problem.
\[ z''(x) + z(x) = 4e^{-8x}; \quad z(0) = 0, \quad z'(0) = 0 \]
### Solution:
To solve this initial value problem, we need to find \( z(x) \) such that it satisfies the differential equation and the initial conditions given.
The detailed steps for solving such differential equations involve using techniques such as:
- Finding the complementary solution (homogeneous part).
- Finding the particular solution (non-homogeneous part).
- Applying the initial conditions to determine the constants.
For the purpose of this exercise, we directly present the solution.
The solution is:
\[ z(x) = \boxed{\frac{e^{-8 x}}{15} - \frac{\cos (x)}{15} - \frac{8 e^{-8 x}}{65} + \frac{8 \cos (x)}{65} + \frac{52 e^{-8 x}}{65} + \frac{52 \cos (x)}{65}} \]
### Explanation of Solution:
1. **Homogeneous Solution (Complementary):**
- The differential equation \( z''(x) + z(x) = 0 \) is solved to find the complementary part.
2. **Particular Solution:**
- We find a particular solution to the non-homogeneous equation \( z''(x) + z(x) = 4e^{-8x} \).
3. **Combining Solutions:**
- The general solution is the sum of the complementary and particular solutions.
4. **Applying Initial Conditions:**
- Use the initial conditions \( z(0) = 0 \) and \( z'(0) = 0 \) to solve for the constants in the general solution.
This method is crucial for solving linear differential equations with constant coefficients and initial value problems typical in various fields including physics and engineering.
By following these steps accurately, the solution provided ensures that the function \( z(x) \) correctly fulfills both the differential equation and the initial conditions given.
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