Put the equation into the form x'(t) = g(t)h(x), and solve by the method of separation of variables. (Hint: factor.) x' = 3-tx² - t + 3x² |x(t) =tan(31 - 2 + c)
Put the equation into the form x'(t) = g(t)h(x), and solve by the method of separation of variables. (Hint: factor.) x' = 3-tx² - t + 3x² |x(t) =tan(31 - 2 + c)
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...
Related questions
Question
![### Problem Statement
**Objective:** Transform the given differential equation into the required form and solve it using the method of separation of variables.
**Equation to Transform:**
\[ x'(t) = 3 - tx^2 - t + 3x^2 \]
**Solution to Verify:**
\[ x(t) = \tan\left(3t - \frac{t^2}{2} + C\right) \]
#### Instructions:
1. **Transformation:** Reorganize the given differential equation into the form \( x'(t) = g(t)h(x) \).
2. **Method:** Apply the method of separation of variables to solve the equation.
### Given Differential Equation
\[ x'(t) = 3 - tx^2 - t + 3x^2 \]
### Proposed Solution
\[ x(t) = \tan\left( 3t - \frac{t^2}{2} + C \right) \]
### Steps for Verification
1. **Transform the Differential Equation:**
- Combine like terms:
\[ x'(t) = (3 - t) + (3x^2 - tx^2) \]
- Factor:
\[ x'(t) = (3 - t)(1 + x^2) \]
2. **Separate Variables and Integrate:**
- Separate \(t\) and \(x\):
\[ \frac{dx}{1 + x^2} = (3 - t) dt \]
- Integrate both sides:
\[ \int \frac{dx}{1 + x^2} = \int (3 - t) dt \]
- Use standard integrals:
\[ \tan^{-1} (x) = 3t - \frac{t^2}{2} + C \]
- Solve for \(x\):
\[ x = \tan \left(3t - \frac{t^2}{2} + C \right) \]
### Conclusion
The proposed solution \( x(t) = \tan \left( 3t - \frac{t^2}{2} + C \right) \) correctly satisfies the transformed differential equation.
This stepwise approach demonstrates the method of separation of variables applied to the given differential equation. Valuable for educational purposes, this solution can help enhance understanding of solving differential equations by](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F004f485a-dc8e-4d2a-a997-494cbc4bff99%2Fbaf48d0e-33a3-4655-9ac9-73f7c2ff5690%2Fp3fes0p_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Objective:** Transform the given differential equation into the required form and solve it using the method of separation of variables.
**Equation to Transform:**
\[ x'(t) = 3 - tx^2 - t + 3x^2 \]
**Solution to Verify:**
\[ x(t) = \tan\left(3t - \frac{t^2}{2} + C\right) \]
#### Instructions:
1. **Transformation:** Reorganize the given differential equation into the form \( x'(t) = g(t)h(x) \).
2. **Method:** Apply the method of separation of variables to solve the equation.
### Given Differential Equation
\[ x'(t) = 3 - tx^2 - t + 3x^2 \]
### Proposed Solution
\[ x(t) = \tan\left( 3t - \frac{t^2}{2} + C \right) \]
### Steps for Verification
1. **Transform the Differential Equation:**
- Combine like terms:
\[ x'(t) = (3 - t) + (3x^2 - tx^2) \]
- Factor:
\[ x'(t) = (3 - t)(1 + x^2) \]
2. **Separate Variables and Integrate:**
- Separate \(t\) and \(x\):
\[ \frac{dx}{1 + x^2} = (3 - t) dt \]
- Integrate both sides:
\[ \int \frac{dx}{1 + x^2} = \int (3 - t) dt \]
- Use standard integrals:
\[ \tan^{-1} (x) = 3t - \frac{t^2}{2} + C \]
- Solve for \(x\):
\[ x = \tan \left(3t - \frac{t^2}{2} + C \right) \]
### Conclusion
The proposed solution \( x(t) = \tan \left( 3t - \frac{t^2}{2} + C \right) \) correctly satisfies the transformed differential equation.
This stepwise approach demonstrates the method of separation of variables applied to the given differential equation. Valuable for educational purposes, this solution can help enhance understanding of solving differential equations by
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning