Question

Solve the initial value problem $$ \frac{d P}{d t}=r\left(1-\frac{P}{P_{e}}\right) P, \quad P(0)=P_{0} $$ by viewing the differential equation as a Bernoulli equation.

Short answer

Question: Solve the initial value problem involving a Bernoulli differential equation, given by: $$ \frac{d P}{d t}=r\left(1-\frac{P}{P_{e}}\right) P, \quad P(0) = P_0 $$ Answer: The solution to the given initial value problem is: $$ P(t) = \frac{1}{-rt + \frac{1}{P_0}} $$

Step-by-step solution

Identify the Bernoulli equation

The given differential equation is: $$ \frac{d P}{d t}=r\left(1-\frac{P}{P_{e}}\right) P $$ This is a Bernoulli equation with n=-1.

Apply substitution method

Let: $$ u=\frac{1}{P} $$ Now we compute \(\frac{du}{dt}\): $$ \frac{d u}{d t} = -\frac{1}{P^2}\frac{d P}{d t} $$ Substitute the given differential equation into this expression: $$ \frac{d u}{d t} = -\frac{1}{P^2}(r\left(1-\frac{P}{P_{e}}\right) P) = -r \left(1 - \frac{P}{P_e}\right) $$ Notice that we now have a linear differential equation in \(u\).

Solve the linear differential equation

The linear differential equation we need to solve is: $$ \frac{d u}{d t} = -r \left(1 - \frac{P}{P_e}\right) $$ Now substitute back using \(u=\frac{1}{P}\) and solve for P: $$ \frac{d\left(\frac{1}{P}\right)}{d t}=-r\left(1-\frac{P}{P_e}\right) \Rightarrow \frac{-d P}{P^2} = -r\left(1 - \frac{P}{P_e}\right) dt $$ Now integrate both sides: $$ \int \frac{-d P}{P^2} = \int -r\left(1 - \frac{P}{P_e}\right) dt $$ This leads to: $$ \frac{1}{P} = -rt + C $$

Use initial condition

Now we use the initial condition \(P(0) = P_0\) to find the constant C: $$ \frac{1}{P_0} = -r(0) + C \Rightarrow C = \frac{1}{P_0} $$ So we have: $$ \frac{1}{P} = -rt + \frac{1}{P_0} $$

Solve for P(t)

Finally, we solve for \(P(t)\): $$ P(t) = \frac{1}{-rt + \frac{1}{P_0}} $$ This is the solution to the initial value problem.

Key concepts

Initial Value Problem

An initial value problem is a type of differential equation paired with a specific initial condition. This means, at the start, or at "time zero" (like when t = 0), the system begins with a known value, given as \( P(0) = P_0 \) in our equation. It's like setting the starting line for an experiment.

• The differential equation describes how variables change, like how the population grows.
• The initial condition provides the starting point, telling us the initial population, \( P_0 \).
• This helps us predict future values because knowing where you start helps you figure out where you’ll go.

Overall, solving such problems helps model real-world phenomena by making calculations relevant to a specific context.

Differential Equations

Differential equations like \( \frac{dP}{dt} = r\left(1-\frac{P}{P_e}\right)P \) describe relationships involving rates of change. They play a crucial role in physics, engineering, and economics.

• They express how a quantity changes over time, such as a population or speed.
• The left side, \( \frac{dP}{dt} \), signifies how a variable, P, changes as time (t) passes.
• The right side defines the relationship causing this change, such as environmental factors or growth rates, here modified by one less the fraction of P over carrying capacity \(P_e\).

Solving these equations helps to predict behavior under various conditions, providing insights into the dynamics of the system being studied.

Substitution Method

The substitution method is a handy technique for simplifying complex differential equations, like changing variables to make equations easier to handle. In this problem, we set \( u = \frac{1}{P} \) to transform our Bernoulli equation into a more manageable form.

• This helps rewrite the equation, turning something cumbersome into something linear and straightforward.
• By substituting, we convert difficult non-linear forms, like when working with \( \frac{dP}{dt} = r\left(1-\frac{P}{P_e}\right)P \), into linear equations in terms of another variable (u).
• The process simplifies solving once we have an equation we can manage more directly.

This approach is like changing the lens's focus to better analyze and solve the equation.

Linear Differential Equation

Linear differential equations are the simpler siblings of more complex types because they define a direct proportionality relationship involving derivatives. Once we applied the substitution \( u = \frac{1}{P} \), the Bernoulli equation transformed.

• In our case, the linear equation is \( \frac{du}{dt} = -r \left(1 - \frac{P}{P_e}\right) \).
• This type of equation is easier to solve because everything is "in a straight line," or simply proportional without powers or complicated terms.
• Linear equations allow straightforward integration or other solving methods, helping to achieve the final solution for \(P(t)\).

Understanding these properties makes tackling these equations less daunting, ensuring you can handle them with ease.