(a) For what values of r does the function y = ex satisfy the differential equation 7y"+ 20y' - 3y = 0? (Enter your answers as a comma-separated list.) r= (b) If r₁ and r₂ are the values of r that you found in part (a), show that every member of the family of functions y = ae 1x + be 2* is also a solution. Let r₁ be the larger value and r₂ be the smaller value. We need to show that every member of the family of functions y = ae 1x + be 2* is a solution of the differential equation 7y" + 20y' - 3y = 0. We must decide whether 7f"(x) + 20f'(x) - 3f(x) = 0 for f(x) = ae 1x + be 2*. Substitute your values for ₁ and ₂ into f(x), then find f'(x) and f"(x). We have f'(x) = and f"(x) = Substituting and combining like terms, we get the following. 7f"(x) +20f'(x) - 3f(x) = 1₁²₁¹x + ( [ Jerzx Therefore, f(x) = ae1* + be 2* is a solution to the differential equation 7y" + 20y' - 3y = 0.
(a) For what values of r does the function y = ex satisfy the differential equation 7y"+ 20y' - 3y = 0? (Enter your answers as a comma-separated list.) r= (b) If r₁ and r₂ are the values of r that you found in part (a), show that every member of the family of functions y = ae 1x + be 2* is also a solution. Let r₁ be the larger value and r₂ be the smaller value. We need to show that every member of the family of functions y = ae 1x + be 2* is a solution of the differential equation 7y" + 20y' - 3y = 0. We must decide whether 7f"(x) + 20f'(x) - 3f(x) = 0 for f(x) = ae 1x + be 2*. Substitute your values for ₁ and ₂ into f(x), then find f'(x) and f"(x). We have f'(x) = and f"(x) = Substituting and combining like terms, we get the following. 7f"(x) +20f'(x) - 3f(x) = 1₁²₁¹x + ( [ Jerzx Therefore, f(x) = ae1* + be 2* is a solution to the differential equation 7y" + 20y' - 3y = 0.
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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## Solving Differential Equations with Exponential Functions
### Problem Statement
**(a)** For what values of \( r \) does the function \( y = e^{rx} \) satisfy the differential equation \( 7y'' + 20y' - 3y = 0 \)? (Enter your answers as a comma-separated list.)
\[ r = \boxed{} \]
**(b)** If \( r_1 \) and \( r_2 \) are the values of \( r \) that you found in part (a), show that every member of the family of functions \( y = ae^{r_1 x} + be^{r_2 x} \) is also a solution.
Let \( r_1 \) be the larger value and \( r_2 \) be the smaller value. We need to show that every member of the family of functions \( y = ae^{r_1 x} + be^{r_2 x} \) is a solution of the differential equation \( 7y'' + 20y' - 3y = 0 \).
- We must decide whether \( 7f''(x) + 20f'(x) - 3f(x) = 0 \) for \( f(x) = ae^{r_1 x} + be^{r_2 x} \).
- Substitute your values for \( r_1 \) and \( r_2 \) into \( f(x) \), then find \( f'(x) \) and \( f''(x) \).
\[
\text{We have } f'(x) = \boxed{} \quad \text{and } f''(x) = \boxed{}.
\]
- Substituting and combining like terms, we get the following:
\[
7f''(x) + 20f'(x) - 3f(x) = \left( \boxed{} \right) e^{r_1 x} + \left( \boxed{} \right) e^{r_2 x}
\]
Therefore, \( f(x) = ae^{r_1 x} + be^{r_2 x} \) is a solution to the differential equation \( 7y'' + 20y' - 3y = 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4acda8c7-3ed9-4816-93cd-e9a52e0b4613%2F61cc8c28-35db-42e9-8650-d1da331d17e4%2Fcygdzb6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Below is the transcription and detailed explanation for an educational website:
---
## Solving Differential Equations with Exponential Functions
### Problem Statement
**(a)** For what values of \( r \) does the function \( y = e^{rx} \) satisfy the differential equation \( 7y'' + 20y' - 3y = 0 \)? (Enter your answers as a comma-separated list.)
\[ r = \boxed{} \]
**(b)** If \( r_1 \) and \( r_2 \) are the values of \( r \) that you found in part (a), show that every member of the family of functions \( y = ae^{r_1 x} + be^{r_2 x} \) is also a solution.
Let \( r_1 \) be the larger value and \( r_2 \) be the smaller value. We need to show that every member of the family of functions \( y = ae^{r_1 x} + be^{r_2 x} \) is a solution of the differential equation \( 7y'' + 20y' - 3y = 0 \).
- We must decide whether \( 7f''(x) + 20f'(x) - 3f(x) = 0 \) for \( f(x) = ae^{r_1 x} + be^{r_2 x} \).
- Substitute your values for \( r_1 \) and \( r_2 \) into \( f(x) \), then find \( f'(x) \) and \( f''(x) \).
\[
\text{We have } f'(x) = \boxed{} \quad \text{and } f''(x) = \boxed{}.
\]
- Substituting and combining like terms, we get the following:
\[
7f''(x) + 20f'(x) - 3f(x) = \left( \boxed{} \right) e^{r_1 x} + \left( \boxed{} \right) e^{r_2 x}
\]
Therefore, \( f(x) = ae^{r_1 x} + be^{r_2 x} \) is a solution to the differential equation \( 7y'' + 20y' - 3y = 0
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