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Give the substitution that converts the given nonlinear equation into a linear equation then find the general solution y'-y=-y³

Give the substitution that converts the given nonlinear equation into a linear equation then find the general solution y'-y=-y³

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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**Converting a Nonlinear Equation into a Linear Equation** *Problem Statement:* Give the substitution that converts the given nonlinear equation into a linear equation, then find the general solution. \[ y' - y = -y^3 \] *Explanation:* 1. **Substitution:** - Find an appropriate substitution to linearize the equation. A common substitution for this type of equation is \( v = y^{-2} \), which simplifies the algebra involved. 2. **Transformation:** - Differentiate \( v = y^{-2} \) with respect to the appropriate variable: \[ \frac{dv}{dt} = -2y^{-3}y' \] - Use this to substitute and transform the equation into a linear form. 3. **Linear Equation:** - Substitute back to form the linear differential equation: \[ \frac{1}{2} \frac{dv}{dt} + v = 1 \] 4. **General Solution:** - Solve the linear differential equation using standard techniques such as an integrating factor or other methods suitable for first-order linear differential equations. 5. **Result:** - Find the general solution in terms of \( y \). This approach demonstrates the method of transforming nonlinear differential equations into linear ones to make them more manageable and solvable.
Transcribed Image Text:**Converting a Nonlinear Equation into a Linear Equation** *Problem Statement:* Give the substitution that converts the given nonlinear equation into a linear equation, then find the general solution. \[ y' - y = -y^3 \] *Explanation:* 1. **Substitution:** - Find an appropriate substitution to linearize the equation. A common substitution for this type of equation is \( v = y^{-2} \), which simplifies the algebra involved. 2. **Transformation:** - Differentiate \( v = y^{-2} \) with respect to the appropriate variable: \[ \frac{dv}{dt} = -2y^{-3}y' \] - Use this to substitute and transform the equation into a linear form. 3. **Linear Equation:** - Substitute back to form the linear differential equation: \[ \frac{1}{2} \frac{dv}{dt} + v = 1 \] 4. **General Solution:** - Solve the linear differential equation using standard techniques such as an integrating factor or other methods suitable for first-order linear differential equations. 5. **Result:** - Find the general solution in terms of \( y \). This approach demonstrates the method of transforming nonlinear differential equations into linear ones to make them more manageable and solvable.
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