Problem 6 - Calculus for Biology and Medicine

Question

Find the equilibria of the following differential equations. $$ \frac{d y}{d t}=\frac{y-1}{y^{2}+1} $$

Step-by-Step Solution

Answer

The equilibrium point is \( y = 1 \).

Step 1: Define Equilibrium Condition

The equilibrium point of a differential equation occurs when the derivative is zero, i.e., \( \frac{dy}{dt} = 0 \). In this case, set the expression equal to zero: \( \frac{y-1}{y^2+1} = 0 \).

Step 2: Solve for y

To solve \( \frac{y-1}{y^2+1} = 0 \), realize that a rational function is zero when its numerator is zero. Therefore, set the numerator equal to zero: \( y - 1 = 0 \).

Step 3: Find Equilibrium Value

Solve the equation from the previous step: \( y - 1 = 0 \) gives \( y = 1 \). This is the equilibrium point of the differential equation.

Key Concepts

Understanding Equilibrium Solutions

An equilibrium solution in a differential equation is a constant solution where the rate of change is zero. In mathematical terms, this means that the derivative of the function is equal to zero. Equilibrium solutions are important because they often represent stable states or steady states that the system described by the differential equation might tend to settle into.

• If you think about it like a seesaw, the equilibrium point is that perfect spot where the seesaw is balanced, neither side moving up or down.
• For a function \(y(t)\), equilibrium points occur when \(\frac{dy}{dt} = 0\).
• To find these points, you solve the equation after setting the derivative to zero.

In the given differential equation, \(\frac{dy}{dt} = \frac{y-1}{y^2+1}\), we are asked to find where this fraction is zero. The balance occurs when the numerator of this rational function is zero, since a zero numerator makes the entire fraction zero. This makes it an equilibrium solution.

Rational Functions Explained

A rational function is any function that can be expressed as the quotient of two polynomial functions. Think of it like a fraction where the top and bottom parts are polynomials. These functions can model many behaviors in fields ranging from physics to economics.

• Every rational function can be written in the form \(f(x) = \frac{p(x)}{q(x)}\), where both \(p(x)\) and \(q(x)\) are polynomials.
• The behavior of a rational function is often determined by its numerator and denominator.
• Understanding when a rational function equals zero is tied to its numerator.

In our scenario, we looked at the rational function \(\frac{y-1}{y^2+1}\). The goal was to find the equilibrium solution by setting the whole expression equal to zero. For rational functions, this is equivalent to setting the numerator \(y-1\) to zero, solving \(y - 1 = 0\), which leads us directly to the answer \(y = 1\).

Basics of Derivatives

Derivatives play a crucial role in the study of differential equations as they fundamentally represent the rate of change. In simple terms, a derivative tells us how a quantity changes with respect to another, offering insights like velocity from position, or growth rates in populations.

• The notation \(\frac{dy}{dt}\) represents the derivative of \(y\) with respect to \(t\).
• When solving differential equations, finding when this derivative is zero helps us identify equilibrium points.
• Generally, if you find \(\frac{dy}{dt} = 0\), this means that \(y(t)\) doesn't change at those specific values of \(t\).

For example, in the differential equation \(\frac{dy}{dt} = \frac{y-1}{y^2+1}\), finding the derivative equal to zero is all about examining when the provided equation stops changing. As explored, this happens when \(y = 1\), giving us a deeper understanding of where these changes cease and a potential system balance could occur.