(a) By inspection, find a particular solution of y" + 2y = 10. Yp(x) = (b) By inspection, find a particular solution of y" + 2y = -4x. Yp(x)= (c) Find a particular solution of y" + 2y = - 4x + 10. y(x) =
(a) By inspection, find a particular solution of y" + 2y = 10. Yp(x) = (b) By inspection, find a particular solution of y" + 2y = -4x. Yp(x)= (c) Find a particular solution of y" + 2y = - 4x + 10. y(x) =
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Problem Set: Finding Particular Solutions to Differential Equations
**(a) By inspection, find a particular solution of:**
\[ y'' + 2y = 10. \]
\[ y_p(x) = \_\_\_\_ \]
**(b) By inspection, find a particular solution of:**
\[ y'' + 2y = -4x. \]
\[ y_p(x) = \_\_\_\_ \]
**(c) Find a particular solution of:**
\[ y'' + 2y = -4x + 10. \]
\[ y_p(x) = \_\_\_\_ \]
**(d) Find a particular solution of:**
\[ y'' + 2y = 8x + 5. \]
\[ y_p(x) = \_\_\_\_ \]
### Explanation
These problems require finding particular solutions to second-order linear differential equations with constant coefficients. Each equation follows the form of:
\[ y'' + 2y = f(x), \]
where \( f(x) \) is a specific function on the right-hand side. The goal is to find solutions that satisfy these equations by inspection or standard methods.
1. **For problem (a):** You need to identify a function \( y_p(x) \) whose second derivative plus twice itself yields 10.
2. **For problem (b):** Identify a function \( y_p(x) \) such that \( y'' + 2y = -4x \).
3. **For problem (c):** Formulate a particular solution for a linear combination on the right-hand side, \( -4x + 10 \).
4. **For problem (d):** Similarly, determine a solution for \( 8x + 5 \).
These exercises develop skills in solving non-homogeneous differential equations using particular solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61a624e8-9805-4882-bd7c-8b439f9ba0a9%2F5fa6835d-7303-487a-a76f-9063f06d9d38%2Ftc95818_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Set: Finding Particular Solutions to Differential Equations
**(a) By inspection, find a particular solution of:**
\[ y'' + 2y = 10. \]
\[ y_p(x) = \_\_\_\_ \]
**(b) By inspection, find a particular solution of:**
\[ y'' + 2y = -4x. \]
\[ y_p(x) = \_\_\_\_ \]
**(c) Find a particular solution of:**
\[ y'' + 2y = -4x + 10. \]
\[ y_p(x) = \_\_\_\_ \]
**(d) Find a particular solution of:**
\[ y'' + 2y = 8x + 5. \]
\[ y_p(x) = \_\_\_\_ \]
### Explanation
These problems require finding particular solutions to second-order linear differential equations with constant coefficients. Each equation follows the form of:
\[ y'' + 2y = f(x), \]
where \( f(x) \) is a specific function on the right-hand side. The goal is to find solutions that satisfy these equations by inspection or standard methods.
1. **For problem (a):** You need to identify a function \( y_p(x) \) whose second derivative plus twice itself yields 10.
2. **For problem (b):** Identify a function \( y_p(x) \) such that \( y'' + 2y = -4x \).
3. **For problem (c):** Formulate a particular solution for a linear combination on the right-hand side, \( -4x + 10 \).
4. **For problem (d):** Similarly, determine a solution for \( 8x + 5 \).
These exercises develop skills in solving non-homogeneous differential equations using particular solutions.
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