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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
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 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
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
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