Problem 9 - Advanced Engineering Mathematics

Question

In Problems, find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=3 x^{2}-4 y \\ &y^{\prime}=x-y \end{aligned} $$

Step-by-Step Solution

Answer

The critical points are \((0, 0)\) and \(\left(\frac{4}{3}, \frac{4}{3}\right)\).

Step 1: Understanding Critical Points

A critical point of a system of differential equations is where the derivatives \(x'\) and \(y'\) are both equal to zero. So, we need to solve the system: \(3x^{2} - 4y = 0\) and \(x - y = 0\).

Step 2: Solve \(x - y = 0\)

The equation \(x - y = 0\) simplifies to \(x = y\). This means that for any critical point, the x-coordinate and y-coordinate must be equal.

Step 3: Substitute \(x = y\) into \(3x^{2} - 4y = 0\)

Substitute \(y = x\) into the first equation: \(3x^{2} - 4(x) = 0\). This simplifies to \(3x^{2} - 4x = 0\).

Step 4: Factor the Simplified Equation

Factor the equation \(3x^{2} - 4x = 0\) to \(x(3x - 4) = 0\). Set each factor to zero: \(x = 0\) and \(3x - 4 = 0\).

Step 5: Solve for x

Solving for \(x = 0\) gives \(x = 0\). Solving \(3x - 4 = 0\) gives \(x = \frac{4}{3}\).

Step 6: Find Corresponding y-values

Since \(x = y\), for each \(x\) value, \(y\) must equal \(x\). So the corresponding \(y\)-values are \(y = 0\) and \(y = \frac{4}{3}\).

Step 7: Identify Critical Points

The critical points are \((0, 0)\) and \(\left(\frac{4}{3}, \frac{4}{3}\right)\).

Key Concepts

Critical Points

In mathematics, particularly in differential equations, critical points play a vital role. They signify locations where a system does not change in time. Thus, to find critical points of a system like our given plane autonomous system, we need to determine where both derivatives become zero. This means solving the equations where the rate of change in both directions (x' and y') is zero. In our case, it involves solving the system:

• For the equation \(3x^{2} - 4y = 0\), any point where the left-hand side equals zero while x and y maintain a relation will be critical.
• For \(x - y = 0\), it simply means x equals y, indicating their equality at any critical point.

By understanding these fundamentals, we can systematically find the critical points where the system reaches equilibrium.

Plane Autonomous System

A plane autonomous system is a type of differential equation where both equations are independent of the variable representing time. That means, in general, the behavior of the system in the plane is solely determined by the interaction of the variables \(x\) and \(y\).

• An autonomous system is typically given in the form of \(x' = f(x, y)\) and \(y' = g(x, y)\), where the functions f and g depend only on the variables x and y.
• In the given example, the system is autonomous since there is no explicit time component, emphasizing the direct relationship between \(x\) and \(y\).

This means the location and nature of critical points stem entirely from internal relationships rather than evolving against an external factor like time.

System of Equations

Systems of equations are collections of equations that must be solved together because they share variables. In differential equations like the given autonomous system, solving a system of equations helps find points where two different expressions both satisfy being zero.

• Our system \(x - y = 0\) and \(3x^{2} - 4y = 0\) needs simultaneous solutions.
• Identifying one variable in terms of another (e.g., \(x = y\)) can simplify solving the system.
• Upon substitution into the second equation, shared solutions emerge, highlighting crucial points where the system stabilizes.

Ultimately, systems of equations in a differential setting help unravel the dynamics and equilibria inherent in interconnected relations.

Factorization

Factorization simplifies solving equations by breaking a complex equation into simpler components. In the context of differential equations, especially for finding critical points, factorization makes solving polynomial equations straightforward.

• For our problem, consider the equation \(3x^{2} - 4x = 0\). By factoring it into \(x(3x - 4) = 0\), the more manageable components \(x = 0\) and \(3x - 4 = 0\) emerge.
• Solving each part independently provides potential x-values crucial for determining critical points.

Through factorization, complex relationships reduce to simple root-finding, enabling swift identification of solutions relevant to the problem at hand.