Solving Bernoulli equations Use the method outlined in Exercise 43 to solve the following Bernoulli equations.


a. y′(t) + y = 2y²

Verified step by step guidance

1. Identify the Bernoulli equation in the form \(y'(t) + P(t)y = Q(t)y^n\). Here, the equation is \(y'(t) + y = 2y^2\), so \(P(t) = 1\), \(Q(t) = 2\), and \(n = 2\).
2. Divide the entire equation by \(y^n = y^2\) (assuming \(y \neq 0\)) to rewrite it as \(y'(t) y^{-2} + y y^{-2} = 2\), which simplifies to \(y'(t) y^{-2} + y^{-1} = 2\).
3. Make the substitution \(z = y^{1-n} = y^{1-2} = y^{-1}\). Then, compute \(z'(t)\) in terms of \(y'(t)\): since \(z = y^{-1}\), we have \(z' = -y^{-2} y'\).
4. Rewrite the original equation in terms of \(z\) and \(z'\). Using \(z' = -y^{-2} y'\), rearrange to express \(y'(t) y^{-2} = -z'\). Substitute into the equation from step 2 to get \(-z' + z = 2\).
5. Rearrange the equation to the linear form \(z' - z = -2\). Solve this first-order linear differential equation for \(z(t)\) using an integrating factor, then substitute back \(y = z^{-1}\) to find the solution for \(y(t)\).

Key Concepts

Bernoulli Differential Equation

A Bernoulli differential equation is a first-order nonlinear ODE of the form y' + P(x)y = Q(x)y^n, where n ≠ 0 or 1. It can be transformed into a linear differential equation by an appropriate substitution, making it easier to solve.

Substitution Method for Bernoulli Equations

To solve a Bernoulli equation, use the substitution v = y^(1-n), which converts the nonlinear equation into a linear one in terms of v. This allows the use of standard methods for linear ODEs to find the solution.

Solving Linear First-Order Differential Equations

Once transformed, the equation becomes linear and can be solved using an integrating factor. The integrating factor is typically e^(∫P(x)dx), which simplifies the equation to an exact derivative, enabling integration and solution.