Problem 11 - Advanced Engineering Mathematics

Question

Solve the following nonlinear plane autonomous system by switching to polar coordinates, and describe the geometric behavior of the solution that satisfies the given initial condition. $$ \begin{aligned} &x^{\prime}=-y-x\left(\sqrt{x^{2}+y^{2}}\right)^{3} \\ &y^{\prime}=x-y\left(\sqrt{x^{2}+y^{2}}\right)^{3}, \quad \mathbf{X}(0)=(1,0) \end{aligned} $$

Step-by-Step Solution

Answer

The solution spirals inward clockwise towards the origin.

Step 1: Convert to Polar Coordinates

To convert the system to polar coordinates, substitute \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). The new variables are \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(\frac{y}{x})\). The system of equations becomes:\[ r' = -r^4 \]\[ \theta' = 1 \]

Step 2: Solve the Radius Equation

The radial equation is \(r' = -r^4\). This is a separable differential equation. Separate the variables to get:\[ \frac{dr}{r^4} = -dt \]Integrate both sides:\[ \int r^{-4} dr = \int -1 dt \]Solving this gives:\[ -\frac{1}{3r^3} = -t + C \]Simplifying, we get:\[ \frac{1}{3r^3} = t + C \]To find \(C\), use the initial condition \(r(0) = 1\):\[ \frac{1}{3(1)^3} = 0 + C \Rightarrow C = \frac{1}{3} \]

Step 3: Solve the Angle Equation

The angular equation is \(\theta' = 1\). This is a simple separable equation:\[ d\theta = dt \]Integrate both sides:\[ \theta = t + C_1 \]Using the initial condition \(\theta(0) = 0\), we find \(C_1 = 0\). Thus, \(\theta(t) = t\).

Step 4: Analyze Geometric Behavior

The solution describes a spiral. The radius \(r(t) = \left(\frac{1}{3(t + \frac{1}{3})}\right)^{1/3}\) decreases over time, while the angle \(\theta(t) = t\) increases, indicating the trajectory spirals inward in a clockwise direction. As \(t \to \infty\), \(r(t) \to 0\) and the trajectory approaches the origin.

Key Concepts

Nonlinear Plane Autonomous System

A nonlinear plane autonomous system is a set of differential equations where the derivatives of certain functions are defined by the functions themselves. These systems are often found in physics and engineering contexts because of their ability to model complex, dynamic behavior over time.
In our example, the system is expressed by the pair of equations \[ x' = -y - x(x^2 + y^2)^{3/2} \] \[ y' = x - y(x^2 + y^2)^{3/2} \]which describe how the rates of change of \(x\) and \(y\) depend on both coordinates as a nonlinear function.

• "Plane" indicates the system operates in the two-dimensional space.
• "Autonomous" means the system's behavior does not explicitly depend on time.
• "Nonlinear" conveys that the relationship between the variables is not merely a simple linear equation.

The use of nonlinear systems allows for the exploration of rich and intricate dynamics, highlighting their significance in studying real-world phenomena.

Geometric Behavior

Geometric behavior in the context of differential equations refers to the nature of the trajectories or paths determined by solutions in the phase plane. Specifically, it tells us how the solutions behave or "look" as time progresses.
In our solved system, we observe that the trajectory traced by the solution is a spiral that moves inward towards the origin. This movement is characterized by two dynamics:

• The radius \(r(t)\) tends to zero as time \(t\) increases, indicating that the spiral tightens around the origin.
• The angle \(\theta(t)\) increases linearly with time, making the spiral expand outward in a rotational sense.

This inward, circular motion signifies that any initial condition set in this system will ultimately approach the origin over time, creating a pattern that is commonly observed in dissipative systems where energy or force gradually diminishes. Understanding the geometric behavior helps us visualize the system's dynamics and predict how the system will evolve, which is crucial for applications in engineering and natural sciences.

Separable Differential Equations

Separable differential equations are a class of differential equations in which the variables can be separated on opposite sides of the equation. This property makes them easier to solve as compared to other types of differential equations.
For the radial component \(r' = -r^4\) in our exercise, we can rewrite it into a separable form: \[ \frac{dr}{r^4} = -dt \]which allows the integration of both sides separately:\[ \int r^{-4} dr = \int -1 dt \]Through integration, we solve:\[ -\frac{1}{3r^3} = -t + C \]Here,

• "Separable" indicates that the differential equation can be reorganized to isolate \(r\) and \(t\) on different sides.
• It simplifies solving complex equations by breaking them into manageable integrals.

Separable differential equations are vital to understanding complex systems as they bridge simple algebraic manipulation with fundamental integration techniques, making them highly useful in various scientific calculations and predictions.