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


c. y′(t) + y = √y

Verified step by step guidance

1. Identify the given differential equation: \(y'(t) + y = \sqrt{y}\). Notice that this is a Bernoulli equation because it can be written in the form \(y' + P(t)y = Q(t)y^n\) with \(n = \frac{1}{2}\).
2. Rewrite the equation explicitly as \(y' + y = y^{1/2}\). Here, \(P(t) = 1\), \(Q(t) = 1\), and \(n = \frac{1}{2}\).
3. Make the substitution \(z = y^{1 - n} = y^{1 - \frac{1}{2}} = y^{\frac{1}{2}}\). This substitution transforms the nonlinear equation into a linear one in terms of \(z\).
4. Differentiate \(z\) with respect to \(t\): \(\frac{dz}{dt} = \frac{1}{2} y^{-\frac{1}{2}} y'\). Solve this expression for \(y'\) to substitute back into the original equation.
5. Substitute \(y'\) in the original equation with the expression in terms of \(z\) and \(\frac{dz}{dt}\), then simplify to get a linear differential equation in \(z\). Solve this linear equation using an integrating factor.

Key Concepts

Bernoulli Differential Equation

A Bernoulli differential equation is a nonlinear first-order ODE of the form y' + P(x)y = Q(x)y^n, where n ≠ 0 or 1. It can be transformed into a linear 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 differential equation in terms of v. This allows the use of standard methods for linear ODEs.

Solving Linear First-Order Differential Equations

Once transformed, the equation becomes linear and can be solved using an integrating factor, μ(t) = e^(∫P(t)dt). Multiplying through by μ(t) simplifies the equation, enabling integration and solution for v(t), and then y(t).