Question

Sometimes it is possible to solve a nonlinear equation by making a change of the dependent variable that converts it into a linear equation. The most important such equation has the form $$ y^{\prime}+p(t) y=q(t) y^{n} $$ and is called a Bernoulli equation after Jakob Bernoulli. the given equation is a Bernoulli equation. In each case solve it by using the substitution mentioned in Problem 27(b). \(y^{\prime}=r y-k y^{2}, r>0\) and \(k>0 .\) This equation is important in population dynamics an is discussed in detail in Section 2.5 .

Short answer

Question: Solve the Bernoulli equation \(y^{\prime} = ry - ky^2\) using the substitution mentioned in the corresponding Problem 27(b). Answer: The solution to the given Bernoulli equation is \(y = \frac{1}{\frac{-k}{r} + Ce^{-rt}}\).

Step-by-step solution

Rewrite the equation in the standard Bernoulli form

We are given the equation as $$y^{\prime} = ry - ky^2$$ To rewrite it in the standard Bernoulli equation form, we need to rewrite it as: $$y^{\prime} - ry = - ky^2$$ So, we have \(p(t) = -r\), \(q(t) = -k\), and \(n = 2\).

Apply the substitution

According to the hint of using the substitution mentioned in Problem 27(b), we need to apply the substitution \(z = y^{1-n} = y^{1-2} = y^{-1}\). This means that \(y = z^{-1}\). Now, let's find the derivative of the substitution: $$z^{\prime} = \frac{d}{dt}y^{-1} = -y^{-2} y^{\prime}$$. Now we will replace y and its derivative in the original equation with the transformed variable z: $$-y^{-2} y^{\prime} - (-r)y = - ky^2$$

Simplify the transformed equation

Replacing \(y^{\prime}\) with \(-y^2 z^{\prime}\) and \(y\) with \(z^{-1}\), we have: $$z^{-1} z^{\prime} + rz^{-1} = -kz^2$$ Multiplying the equation by \(z^2\), we get: $$z^{\prime} + r z = -k$$ Now, we have a transformed linear equation that we can solve using an integrating factor.

Solve the linear equation

Let's find an integrating factor for the linear equation with z. The integrating factor is given by: $$\mu(t) = e^{\int rt dt} = e^{r \int t dt} = e^{rt}$$ Now multiply the transformed linear equation by the integrating factor: $$e^{rt} z^{\prime} + r e^{rt} z = - e^{rt} k$$ Now, we can observe that the left side of the equation is the derivative of \((e^{rt} z)\) with respect to t. So, we can write it as: $$\frac{d}{dt}(e^{rt} z) = - e^{rt} k$$ Now, integrate both sides of the equation with respect to t: $$\int \frac{d}{dt}(e^{rt} z) dt = \int - e^{rt} k dt$$ $$e^{rt} z = \frac{- k}{r} e^{rt} + C$$ Now, solve for z: $$z = \frac{- k}{r} + Ce^{-rt}$$

Revert the substitution and obtain the solution for y

Now, we need to revert the substitution. Recall that \(y = z^{-1}\). So, the solution for y is given by: $$y = \frac{1}{z} = \frac{1}{\frac{- k}{r} + Ce^{-rt}}$$ And this is the solution to the given Bernoulli equation.

Key concepts

Nonlinear Differential Equations

Nonlinear differential equations are equations that involve derivatives with respect to a function and where the function or its derivatives are raised to a power other than one. These are important because many natural phenomena are described by such equations. In mathematical terms, a nonlinear differential equation does not satisfy the superposition principle, meaning solutions cannot be simply added together to form new solutions.

The Bernoulli equation is a classic example of a nonlinear differential equation. It includes a term where the dependent variable is raised to a power, which makes the equation nonlinear. Specifically, for the equation \( y^{\prime}+p(t) y=q(t) y^{n} \), it becomes nonlinear due to the term \( q(t) y^n \).

Solving such equations often requires transforming them into linear equations through methods like substitutions. Once converted, they can be tackled using techniques for linear equations, making them easier to solve. Understanding these transformations is crucial for dealing with more complex systems modeled by nonlinear dynamics.

Linearization Techniques

Turning a nonlinear equation into a simpler linear form can be a powerful technique, often referred to as linearization. This is achieved through smart substitutions that simplify the nonlinear terms. In the context of the Bernoulli equation, when faced with \( y^{\prime} = ry - ky^2 \), a substitution like \( z = y^{-1} \) is used. This specific form helps transform the equation into a linear one, which is easier to solve.

The substitution effectively removes the complexity introduced by the non-linear term \( ky^2 \), allowing the equation to adhere to a linear structure. This acts like taking a complex problem and peeling away layers until we have something more manageable.

Once the equation is linearized, integrating factors can be applied, a standard technique for solving linear differential equations. Recognizing when and how to apply substitutions is critical for solving complex non-linear differential equations in fields like physics and population dynamics.

Integrating Factors

When a differential equation is linear, particularly in the form \( y^{\prime} + P(t)y = Q(t) \), integrating factors are a useful method to simplify and solve the equation. An integrating factor is a function, often denoted as \( \mu(t) \), which, when multiplied through the equation, allows the left-hand side to be expressed as the derivative of a product of functions.

In the linearized Bernoulli equation resulting from the substitution \( z = y^{-1} \), the integrating factor \( \mu(t) \) is chosen as \( e^{\int P(t) dt} \). In our case, \( \mu(t) = e^{rt} \). This multiplying factor transforms the equation such that it can be written as \( \frac{d}{dt}(e^{rt} z) = - e^{rt} k \).

Once this transformation is performed, the differential equation can be integrated with respect to \( t \), solving for the transformed variable. Integrating factors offer a systematic way to simplify and solve differential equations that might initially appear challenging.

Population Dynamics

Population dynamics is an interesting field where mathematical models like differential equations are used to describe how populations change over time under various influences. The equation \( y^{\prime} = ry - ky^2 \) is particularly significant in this context. It represents what is known as the logistic growth model, which accounts for limited resources, where \( r \) is the growth rate and \( k \) is the carrying capacity.

In simple terms, the term \( ry \) describes exponential growth of the population without resource limitations, while \( ky^2 \) represents competition within the population for limited resources. This equation predicts that populations will initially grow rapidly, but as resources become scarce, the growth rate will slow and eventually stabilize.

This mathematical model provides valuable insights into how species populations evolve in natural environments and can be critical in fields such as ecology, conservation biology, and resource management. Solving such equations allows us to predict future population sizes and help in strategizing for sustainable development.