Problem 38 - 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. $$ \frac{d y}{d x}=y^{4}+y^{3}-1 $$

Step-by-Step Solution

Answer

Solve \( y^4 + y^3 - 1 = 0 \) for equilibria, then analyze sign changes to classify stability.

Step 1: Understand the Differential Equation

The given differential equation is \( \frac{dy}{dx} = y^4 + y^3 - 1 \). We need to find the equilibria, which occur when the derivative is zero, i.e., \( y^4 + y^3 - 1 = 0 \).

Step 2: Set Up the Equation for Equilibria

To find equilibria, solve the equation \( y^4 + y^3 - 1 = 0 \). These solutions will tell us the values of \( y \) where the slope \( \frac{dy}{dx} \) is zero.

Step 3: Apply Methods to Factor or Solve Equation

This is a polynomial equation, which may need factoring or numerical methods. Begin by checking for rational roots or attempt synthetic division to factor it into simpler components if possible, or solve numerically if necessary.

Step 4: Analyzing Sign Changes

After obtaining the possible real roots, analyze the sign of \( \frac{dy}{dx} = y^4 + y^3 - 1 \) in each interval between the roots to determine the stability. Stable equilibria will show a change from positive to negative as \( y \) crosses from left to right (\( \frac{dy}{dx} \) negative to positive). Unstable will show negative to positive shifts.

Step 5: Classify Each Equilibria

From the sign analysis, classify each equilibrium as stable or unstable by noting the change in signs of \( \frac{dy}{dx} \) around each equilibrium point. Here, if the derivative goes from positive to negative around a root, it is stable; if it goes from negative to positive, it is unstable.

Key Concepts

Equilibria

In the context of differential equations, equilibria refer to the points where the rate of change is zero. When we're asked to find the equilibria of a differential equation like \( \frac{dy}{dx} = y^4 + y^3 - 1 \), we're essentially trying to determine the values of \( y \) that make the entire expression equal to zero. The equation \( y^4 + y^3 - 1 = 0 \) arises from setting the derivative to zero, as this indicates no change in \( y \) over time.

Equilibria are important because they represent steady states where the system does not change unless disturbed by an external force or condition. These points of equilibrium can then be analyzed to determine whether they are stable or unstable, which gives insight into the behavior of the system over time.

Finding differing equilibrium points involves solving polynomial equations, which we will cover in more detail later. Once these points are found, they form the basis for further stability analysis.

Stability Analysis

Stability analysis involves examining how solutions to the differential equation behave as they get close to equilibrium points. Specifically, stability tells us whether small deviations from equilibrium will fade away or amplify over time.

To analyze stability in the given differential equation \( \frac{dy}{dx} = y^4 + y^3 - 1 \), we first locate the equilibrium points by solving the equation \( y^4 + y^3 - 1 = 0 \). Once we have these points, stability is determined by observing the behavior of the derivative on either side of these points.

Consider the following when examining each equilibrium point:

• If \( \frac{dy}{dx} \) changes from positive to negative as \( y \) crosses the equilibrium point from left to right, the equilibrium is stable. Small deviations will cause \( y \) to return to the equilibrium.
• If \( \frac{dy}{dx} \) changes from negative to positive, the equilibrium is unstable. Small deviations will cause \( y \) to move away from the equilibrium.

The direction of these changes can often be determined from the sign of \( \frac{dy}{dx} \) in the regions around each equilibrium point, making sign analysis an essential tool in stability analysis.

Polynomial Equations

Polynomial equations are equations involving variables raised to whole number powers, and they form the backbone of many calculations in differential equations, especially when finding equilibria. The differential equation \( \frac{dy}{dx} = y^4 + y^3 - 1 \) requires solving the associated polynomial equation \( y^4 + y^3 - 1 = 0 \) to find equilibria.

Solving polynomial equations often involves several methods:

• Factoring: Factoring the polynomial into simpler binomial or trinomial forms can reveal solutions directly. Factoring is straightforward for lower degree polynomials but can become complex as the degree increases.
• Rational Root Theorem: This theorem provides possible rational roots based on factors of the constant term and leading coefficient, which can then be used to try to simplify the equation.
• Numerical Methods: For complex polynomials that resist simple algebraic solutions, numerical methods like Newton's method or graphing can approximate the roots.

Understanding and solving polynomial equations is crucial as they pave the way for deeper analysis like stability, crucially answering whether such points are attractors or repellers in the system.