Problem 7 - Calculus for Biology and Medicine

Question

Find the equilibria of the following differential equations. $$ \frac{d x}{d t}=x^{8}-1 $$

Step-by-Step Solution

Answer

The equilibria are at \( x = 1 \) and \( x = -1 \) (real), with additional complex solutions given by eighth roots of unity.

Step 1: Set the Differential Equation to Zero

To find the equilibria, set the differential equation \( \frac{dx}{dt} = x^8 - 1 \) equal to zero, because equilibria occur when the rate of change is zero.

Step 2: Solve the Resulting Algebraic Equation

Solve the equation \( x^8 - 1 = 0 \). This can be rewritten as \( x^8 = 1 \).

Step 3: Find the Real Solutions

To find the real solutions to \( x^8 = 1 \), note that \( x = 1 \) and \( x = -1 \) satisfy this equation, since both elevate to a positive power that results in 1.

Step 4: Consider Complex Solutions

Because \( x^8 = 1 \) is a complex equation too, the complete solution for \( x \) involves all eighth roots of unity, which are complex numbers in the form \( e^{2\pi i k/8} \) for \( k = 0, 1, ..., 7 \).

Key Concepts

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. These equations play a fundamental role in various fields like physics, engineering, and economics. The primary aim of solving differential equations is to find unknown functions that fulfill the given equation. They are broadly classified into two types—ordinary differential equations (ODEs) and partial differential equations (PDEs). While ODEs involve functions of a single variable, PDEs deal with functions of multiple variables. In our case, the equation \( \frac{d x}{d t} = x^8 - 1 \) is an ordinary differential equation. To find the equilibrium points, one sets the rate of change, represented by \( \frac{d x}{d t} \), to zero, because equilibrium points occur where the function is constant, meaning no change over time.

Roots of Unity

Roots of unity are fundamental when dealing with complex numbers and polynomial equations, particularly when these equations have solutions over the complex number field. The \( n \)-th roots of unity are solutions to the equation \( z^n = 1 \), where \( z \) is a complex number. In the context of our differential equation, when set into a form where complex solutions are valid \( x^8 = 1 \), we discover that there are not just real solutions like \( x = 1 \) and \( x = -1 \), but also complex solutions. These solutions can be represented by the formula \( e^{2\pi i k/8} \) for \( k = 0, 1, ..., 7 \). They form a circle in the complex plane, with each one equally spaced around the unit circle centered at the origin. These roots of unity denote the eighth roots of 1, which include the aforementioned real roots as well as other non-real complex numbers.

Stability Analysis

Stability analysis is a method used to determine whether equilibrium points of a differential equation are stable or unstable. This involves analyzing how the system behaves over time in the vicinity of these points. For first-order differential equations like \( \frac{d x}{d t} = x^8 - 1 \), this often involves checking the sign of the derivative around the equilibrium points. If we assess the derivative \( \frac{d^2 x}{d t^2} \) at solutions such as \( x = 1 \) and \( x = -1 \), positive derivatives suggest instability while negative derivatives suggest stability. For complex-valued equilibrium points, the analysis can become more intricate, often involving linear approximations or numerical simulations to predict system behavior. The aim is to understand whether small perturbations around an equilibrium will die out, reinforcing the equilibrium, or grow, leading the system away from equilibrium.