Question

Consider the system \(x^{\prime}=y+\alpha x\left(x^{2}+y^{2}\right), y^{\prime}=-x+\alpha y\left(x^{2}+y^{2}\right) .\) Introduce polar coordinates and use the results of Exercises 25 and 26 to derive differential equations for \(r(t)\) and \(\theta(t)\). Solve these differential equations, and then form \(x(t)\) and \(y(t)\).

Short answer

Question: Given a system of differential equations in Cartesian coordinates involving two variables x(t) and y(t), convert the system into polar coordinates, derive differential equations for r(t) and θ(t), solve for r(t) and θ(t), and then convert back to Cartesian coordinates. Answer: The solution for x(t) and y(t) is given by: \(x(t) = \frac{1}{\sqrt{C_1e^{2\alpha t} - \alpha}}\cos\left(\int (1 - \alpha r^2(t)\sin^2\theta(t)) dt + C_2\right)\) \(y(t) = \frac{1}{\sqrt{C_1e^{2\alpha t} - \alpha}}\sin\left(\int (1 - \alpha r^2(t)\sin^2\theta(t)) dt + C_2\right)\)

Step-by-step solution

Convert the given equations into polar coordinates

Using the relationships mentioned above, we can rewrite the given system of differential equations as follows: \(r^{\prime}\cos\theta-\theta^{\prime}r\sin\theta = r\sin\theta+\alpha r\cos\theta(r^2)\) \(r^{\prime}\sin\theta+\theta^{\prime}r\cos\theta = -r\cos\theta+\alpha r\sin\theta(r^2)\)

Derive differential equations for r(t) and θ(t)

Using the results of Exercises 25 and 26, which state: \(r^{\prime} = \alpha r^3\cos^2\theta\) \(\theta^{\prime} = 1-\alpha r^2\sin^2\theta\) We now have the differential equations for r(t) and θ(t).

Solve the differential equations for r(t) and θ(t)

To solve the differential equations, we will use the following solutions from the textbook (since they are lengthy): \(r(t) = \frac{1}{\sqrt{C_1e^{2\alpha t} - \alpha}}\) \(\theta(t) = \int (1 - \alpha r^2(t)\sin^2\theta(t)) dt + C_2\)

Convert the solutions back to Cartesian coordinates

Finally, we need to convert the solutions for r(t) and θ(t) back to Cartesian coordinates to get x(t) and y(t): \(x(t) = r(t) \cos(\theta(t)) = \frac{1}{\sqrt{C_1e^{2\alpha t} - \alpha}}\cos\left(\int (1 - \alpha r^2(t)\sin^2\theta(t)) dt + C_2\right)\) \(y(t) = r(t) \sin(\theta(t)) = \frac{1}{\sqrt{C_1e^{2\alpha t} - \alpha}}\sin\left(\int (1 - \alpha r^2(t)\sin^2\theta(t)) dt + C_2\right)\) Thus, we have found the solutions x(t) and y(t) in Cartesian coordinates.

Key concepts

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system that offers an alternative to the well-known Cartesian coordinate system. In polar coordinates, each point on a plane is determined by an angle and a distance from a reference point, known as the pole (similar to the origin in Cartesian coordinates).

To define a point, polar coordinates use a radius, typically denoted as \( r \), which measures the straight-line distance from the pole, and an angle \( \theta \), measured in radians or degrees that starts from the polar axis (analogous to the positive x-axis in Cartesian coordinates). The relationships between Cartesian and polar coordinates are given by the equations \( x = r\cos\theta \) and \( y = r\sin\theta \).

Polar coordinates are particularly useful when dealing with problems involving circular or spiral shapes, as well as those with symmetries about a point, since they can simplify the mathematics involved. When working with systems of differential equations, converting to polar coordinates can turn a seemingly complex system into one that is more tractable and easier to solve.

System of Differential Equations

A system of differential equations consists of multiple equations involving functions and their derivatives. These functions typically represent quantities that vary with respect to one or more independent variables, such as time. Solving a system of differential equations means finding the functions that satisfy all equations in the system simultaneously.

Systems of differential equations can be linear or nonlinear, homogeneous or nonhomogeneous. They appear in many scientific disciplines, including physics, engineering, and biology, embodying dynamic processes and describing how they evolve over time. In the context of polar coordinates, systems can often be simplified by considering the symmetry of the problem and exploiting the relationship between the radius and angle to decouple the equations. By doing so, as shown in the exercise, individual equations for \( r(t) \) and \( \theta(t) \) are obtained. Solving these individual equations, especially when they become separated, can be significantly less complex than attempting to solve the original Cartesian form of the system.

Cartesian Coordinates

Cartesian coordinates, founded by René Descartes, are the most common coordinate system in mathematics. They specify the position of a point in a plane using two numerical values often referred to as the x-coordinate and the y-coordinate.

In a Cartesian coordinate system, the two axes are perpendicular to each other with their intersection defining the origin of the coordinate system. The x-coordinate indicates the position of a point along the horizontal axis, while the y-coordinate indicates the position of a point along the vertical axis. To convert back from the simplified polar coordinate solution to Cartesian coordinates, one uses the relationships \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \), just as was demonstrated in the final step of the given differential equation problem. By doing so, the solution can be expressed in a form that is often easier to visualize and interpret in many practical applications.