Problem 32 - Calculus for Biology and Medicine

Question

By breaking down each equation into two parts that you can sketch, determine how many equilib\mathrm{\\{} r i a ~ e a c h ~ d i f f e r e n t i a l ~ e q u a t i o n ~ h a s , ~ a n d ~ c l a s s i f y ~ t h e m ~ a s ~ s t a b l e ~ or unstable. You do not need to determine the location of the equilibria. $$ \frac{d x}{d t}=3 e^{-x^{2}}-x^{2} $$

Step-by-Step Solution

Answer

There are three equilibrium points; classify them by visually examining the intersection slopes: typically, one stable and two unstable.

Step 1: Identify the Given Differential Equation

We are given the differential equation \( \frac{dx}{dt} = 3e^{-x^2} - x^2 \). We need to find its equilibrium points by setting \( \frac{dx}{dt} = 0 \).

Step 2: Set the Differential Equation to Zero

Equilibrium points occur where \( \frac{dx}{dt} = 0 \). Therefore, we set up the equation: \[ 3e^{-x^2} - x^2 = 0 \] and solve for \( x \).

Step 3: Split the Equation into Two Parts

For finding intersections, consider the functions \( y_1 = 3e^{-x^2} \) and \( y_2 = x^2 \). We solve for equilibrium by finding intersections of these two curves.

Step 4: Plot Both Functions To Determine Intersections

Graph both \( y_1 = 3e^{-x^2} \) and \( y_2 = x^2 \). The intersections of these graphs indicate the equilibrium points. Typically, sketching these functions will provide a visual insight into the number of intersections.

Step 5: Determine Stability of Each Equilibrium Point

Stability can be assessed by examining the slopes near the equilibrium points: - If \( \frac{dy}{dx} > 0 \) right after an intersection and \( \frac{dy}{dx} < 0 \) right before, it is stable.- If \( \frac{dy}{dx} < 0 \) right after and \( \frac{dy}{dx} > 0 \) right before, it is unstable.

Key Concepts

Equilibrium Points

An equilibrium point in a differential equation is where the rate of change is zero. For the equation \( \frac{dx}{dt} = 3e^{-x^2} - x^2 \), finding these points is essential as they represent where the system can rest without any external forces causing change.
To find an equilibrium point, we set the rate of change to zero that is, \( \frac{dx}{dt} = 0 \). This simplifies our task to solving the equation \( 3e^{-x^2} - x^2 = 0 \).
The equilibrium points are where the two curves \( y_1 = 3e^{-x^2} \) and \( y_2 = x^2 \) intersect. When these curves meet, the functionality balances out, resulting in no net change.
In practical applications, equilibrium points can tell much more. They often represent stable states of environmental systems, populations, chemical reactions, and many other dynamic systems.

Stability Analysis

Once equilibrium points are identified, we proceed to study their stability. This tells us if the system will return to equilibrium after a disturbance or move away.
In differential equations, stability is usually assessed by examining the slope around the equilibrium point. To check stability, we look at the direction of the function's slope \((\frac{dy}{dx})\) just before and after an equilibrium.

• If \(\frac{dy}{dx} > 0\) or slopes upwards smoothly before the point and \(\frac{dy}{dx} < 0\) after (it slopes downwards), the point is stable. It attracts and restores the system when slightly disturbed.
• Conversely, if the slope shifts from negative before the equilibrium point to positive after, then the point is considered unstable. In this case, the system will diverge away from equilibrium with disturbances.

This analysis can help in predicting the future behavior of a system, shedding light on potential steady states or oscillations inherent in different real-world processes.

Graphical Method

The graphical method involves plotting the functions to visually determine the intersection points, which represent equilibrium. It’s a straightforward approach to finding equilibrium points without solving equations algebraically.
In the equation \( \frac{dx}{dt} = 3e^{-x^2} - x^2 \), the curves \( y_1 = 3e^{-x^2} \) and \( y_2 = x^2 \) are plotted. These plots will intersect at equilibrium points. This method is particularly insightful because:

• It provides a clear visual representation. You can see how the functions behave and identify intersections easily.
• Approximating where curves intercept gives insight into equilibrium numbers and nature. It helps predict whether the equilibria are stable or unstable by observing slopes.

While algebraic solutions offer precision, graphical methods are invaluable for an intuitive understanding and a first-step analysis. Revisiting the graphic plots with this understanding aids in more complex computations and models.