Problem 43 - Advanced Engineering Mathematics

Question

The equations in the system \(\frac{d x}{d t}=\frac{x}{x^{2}+y^{2}}, \quad \frac{d y}{d t}=\frac{y}{x^{2}+y^{2}}\) can be divided to give \(\frac{d y}{d x}=\frac{y}{x}\).By separation of variables we obtain \(y=c x.\)

Step-by-Step Solution

Answer

The solution is \( y = cx \), where \( c \) is a constant.

Step 1: Understand the Given System

The system of equations given is \( \frac{dx}{dt} = \frac{x}{x^2 + y^2} \) and \( \frac{dy}{dt} = \frac{y}{x^2 + y^2} \). These are two differential equations describing the changes in \( x \) and \( y \) with respect to \( t \).

Step 2: Divide the Equations

To eliminate the parameter \( t \), we divide the second equation by the first. This results in \( \frac{dy}{dt} \div \frac{dx}{dt} = \frac{y}{x} \). Thus, the equation becomes \( \frac{dy}{dx} = \frac{y}{x} \).

Step 3: Apply Separation of Variables

We separate variables in \( \frac{dy}{dx} = \frac{y}{x} \) by rearranging terms to \( \frac{dy}{y} = \frac{dx}{x} \). This allows us to integrate both sides separately.

Step 4: Integrate Both Sides

Integrate \( \int \frac{dy}{y} = \int \frac{dx}{x} \). The left side integrates to \( \ln |y| \) and the right side integrates to \( \ln |x| \). Thus, we have \( \ln |y| = \ln |x| + C \), where \( C \) is the constant of integration.

Step 5: Exponentiate to Solve for y

Exponentiate both sides of the equation \( \ln |y| = \ln |x| + C \) to solve for \( y \). This gives \( |y| = e^C |x| \), which we can write as \( y = cx \) where \( c = e^C \) is a constant.

Key Concepts

System of Equations

When you have multiple equations with more than one unknown, you're working with a system of equations. Systems of equations can be complex because they involve solving not one, but usually two or more equations simultaneously. In our problem, we have two equations:

• \( \frac{dx}{dt} = \frac{x}{x^2 + y^2} \)
• \( \frac{dy}{dt} = \frac{y}{x^2 + y^2} \)

These describe how the variables \( x \) and \( y \) change over time \( t \). The goal is often to eliminate the parameter \( t \) and find a direct relationship between \( x \) and \( y \). In this exercise, the system is simplified by dividing the second equation by the first. This approach eliminates \( t \) and allows us to focus on the relationship between \( x \) and \( y \) directly. Understanding and breaking down systems is critical because they appear in many real-world applications, such as physics, engineering, and economics.

Separation of Variables

In differential equations, separating variables is a key method you use to solve equations of the form \( \frac{dy}{dx} = f(x)g(y) \). The goal is to separate all \( y \)'s on one side and \( x \)'s on the other. Here, we've got the equation:

• \( \frac{dy}{dx} = \frac{y}{x} \)

We can rewrite this as \( \frac{dy}{y} = \frac{dx}{x} \). This separation allows us to focus on each variable independently. Once separated, you can move on to the next step, which is integration. Separation of variables is particularly useful because it simplifies complex problems into manageable parts. It's a bit like solving a puzzle, where you're putting similar pieces together before seeing the whole picture.

Integration

Integration is like the reverse of differentiation. When we integrate a function, we find all possible functions (anti-derivatives) whose derivative is the given function. Here, we integrate both sides:

• \( \int \frac{dy}{y} = \int \frac{dx}{x} \)

The result of these integrals are

• \( \ln |y| \)
• \( \ln |x| \)

plus some constant of integration. So, you get the equation \( \ln |y| = \ln |x| + C \). This step is crucial because it allows us to find a general solution to our differential equation. By reversing the differentiation process, integration provides a way to calculate original quantities given the rate of change.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) involve functions of a single variable and their derivatives. They are foundational in understanding how systems change over time. In our exercise, the ODEs are:

• \( \frac{dx}{dt} = \frac{x}{x^2 + y^2} \)
• \( \frac{dy}{dt} = \frac{y}{x^2 + y^2} \)

These ODEs describe how \( x \) and \( y \) evolve as time \( t \) progresses. Solving ODEs typically involves finding a function or functions that satisfy the differential equations. They require special techniques, such as separation of variables or integration, to solve. Understanding ODEs is vital since they model a vast array of scientific phenomena, such as population dynamics, heat transfer, and fluid flow. Their solutions often reveal insights into the behavior of the system being studied.