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Chapter 8: Problem 54

Plot a solution to the initial-value problem $$ \frac{d y}{d t}=0.98\left(1-\frac{y}{5}\right) y, \quad y_{0}=1 $$

Short Answer

Expert verified
The solution to the initial-value problem shows logistic growth with \( y(t) \) approaching 5.

Step by step solution

01

Understand the Differential Equation

The given differential equation is of the form \( \frac{dy}{dt} = f(t, y) \), with \( f(t, y) = 0.98 \left(1 - \frac{y}{5}\right) y \). This represents a first-order non-linear differential equation.
02

Identify the Initial Condition

The initial condition provided is \( y(0) = 1 \). This means that at \( t = 0 \), the value of \( y \) is \( 1 \). This will be used to plot the solution from this initial point.
03

Analyze the Equation

The expression \( 0.98 \left(1 - \frac{y}{5}\right) y \) indicates that the rate of change of \( y \) with respect to \( t \) depends on \( y \) itself. The term \( \left(1 - \frac{y}{5}\right) \) suggests logistic growth, where \( y \) approaches 5 as \( t \) increases.
04

Consider the Qualitative Behavior

As \( y \) increases towards 5, \( \left(1 - \frac{y}{5}\right) \) decreases, slowing the growth of \( y \). With \( y_0 = 1 \), \( y(t) \) should grow exponentially initially, then slow down as it nears \( y = 5 \).
05

Plotting the Solution

Plot the function \( y(t) \) starting from the initial condition \( (0, 1) \). You'll start with a steep slope, then gradually flatten as \( y \) approaches 5. Numerical methods or software (like Python, MATLAB) can verify the exact behavior.
06

Verify against Expected Behavior

Ensure that the plot resembles the logistic growth curve: initially exponential growth that slows and approaches a horizontal asymptote at \( y = 5 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First-Order Differential Equations
First-order differential equations involve derivatives of a function with respect to one variable, typically time, and are of the form \( \frac{dy}{dt} = f(t, y) \). They are fundamental in modeling systems where the rate of change depends only on the current state and time.
  • Key Feature: The dependent variable (like \( y \) here) and its derivative only depend on the present value, not any higher derivatives.
  • Non-linearity: Our equation \( \frac{dy}{dt} = 0.98\left(1 - \frac{y}{5}\right) y \) is non-linear because the dependent variable \( y \) is multiplied by itself.
First-order equations often model real-world phenomena, such as population growth, radioactive decay, and cooling processes. Understanding and solving these equations can give insights into how these processes behave over time.
Logistic Growth
Logistic growth models are popular for describing populations and processes that start with exponential growth and then slow as they reach carrying capacity.
  • Carrying Capacity: In our problem, the term \( \left(1 - \frac{y}{5}\right) \) hints that the carrying capacity is \( 5 \). This means as \( y \) approaches \( 5 \), growth slows down.
  • Growth Rate: Initially, when \( y \) is much smaller than \( 5 \), \( y \) grows rapidly since \( \left(1 - \frac{y}{5}\right) \approx 1 \).
As \( y \) increases, the factor \( \left(1 - \frac{y}{5}\right) \) decreases, illustrating why the growth decelerates. This behavior results in a characteristic "S" shaped curve when plotting \( y(t) \) over time, known as the logistic curve.
Numerical Methods
Numerical methods are tools to approximately solve differential equations when an exact solution is difficult to find.
  • Why Use Numerical Methods: Our equation is non-linear and solving it analytically might not be straightforward.
  • Common Methods: Techniques like Euler's method and Runge-Kutta methods allow us to calculate estimates by incrementing small steps forward in time.
Utilizing software like Python or MATLAB can simplify the implementation of these methods efficiently. Numerical methods are vital in practical applications where modeling complex systems analytically is challenging or impossible. By using these tools, one can generate plots and understand the system's behavior over time, as with plotting \( y(t) \) in the given problem.

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Most popular questions from this chapter

Find a solution to the initial-value problem. \(x y^{\prime}+y=e^{x}, y(1)=1+e \quad\) (See Exercise 25.)

Solve the initial-value problem. \(x \frac{d y}{d x}-y=x^{2}, \quad y(1)=-1\)

Suppose that an initial population of 10,000 bacteria grows exponentially at a rate of \(2 \%\) per hour and that \(y=y(t)\) is the number of bacteria present \(t\) hours later. (a) Find an initial-value problem whose solution is \(y(t)\). (b) Find a formula for \(y(t)\). (c) How long does it take for the initial population of bacteria to double? (d) How long does it take for the population of bacteria to reach \(45,000 ?\)

Consider the initial-value problem $$ \frac{d y}{d x}=\frac{\sqrt{y}}{2}, \quad y(0)=1 $$ (a) Use Euler's Method with step sizes of \(\Delta x=0.2\), \(0.1\), and \(0.05\) to obtain three approximations of \(y(1)\). (b) Find \(y(1)\) exactly.

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of \(x\) $$ y^{\prime}=-x y $$
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