Problem 6 - Advanced Engineering Mathematics

Question

Let \(M=4 x^{3}-3 y \sin 3 x-y / x^{2}\) and \(N=2 y-1 / x+\cos 3 x\) so that \(M_{y}=-3 \sin 3 x-1 / x^{2}\) and \(N_{x}=1 / x^{2}-3 \sin 3 x\) The equation is not exact.

Step-by-Step Solution

Answer

The given equation is not exact because \(M_y \neq N_x\).

Step 1: Understanding Exactness

A differential equation is said to be exact if the partial derivative of \( M \) with respect to \( y \) is equal to the partial derivative of \( N \) with respect to \( x \). This would imply that \( M_y = N_x \).

Step 2: Reviewing Given Derivatives

From the problem statement, we know \( M_y = -3 \sin 3x - \frac{1}{x^2} \) and \( N_x = \frac{1}{x^2} - 3 \sin 3x \).

Step 3: Evaluating Equality

Comparing the derivatives, \( M_y = -3 \sin 3x - \frac{1}{x^2} \) and \( N_x = \frac{1}{x^2} - 3 \sin 3x \), we see that they do not equal (as their components are reversed).

Step 4: Conclusion on Exactness

Since \( M_y \) is not equal to \( N_x \), we conclude that the differential equation is not exact.

Key Concepts

Partial Derivatives

Partial derivatives are foundational in multivariable calculus. They measure the rate at which a function changes as one variable changes, while keeping other variables constant. In our original exercise, we are given functions \(M(x,y)\) and \(N(x,y)\), and we need to find the partial derivatives \(M_y\) and \(N_x\).

• \(M_y\): The partial derivative of \(M\) with respect to \(y\) involves treating \(x\) as a constant and differentiating each term of \(M\) where \(y\) appears.
• \(N_x\): Similarly, \(N_x\) requires differentiating \(N\) with respect to \(x\), treating \(y\) as a constant.

These derivatives help determine if a differential equation is exact. Mastering this concept is crucial as it forms the basis of checking the exactness condition.

Exactness Condition

The exactness condition is a specific criterion used in the analysis of differential equations. A differential equation like \(M(x,y)dx + N(x,y)dy\) is exact when two partial derivatives are equal: the partial derivative of \(M\) with respect to \(y\) and the partial derivative of \(N\) with respect to \(x\).

In other words, we have:

• \(M_y = N_x\)

This condition implies that there exists some potential function \(\Phi(x, y)\) such that \(d\Phi = Mdx + Ndy\). Determining that \(M_y eq N_x\), as in the exercise, signifies the equation is not exact, so a simple potential function does not exist. Understanding this condition helps in finding solutions to exact differential equations more easily.

Differential Equations

Differential equations are equations that relate a function to its derivatives. They are crucial in modeling how quantities change and are applied widely in fields such as physics, engineering, and economics.

• An exact differential equation is a special type that satisfies \(M_y = N_x\).
• If exact, these equations can be solved by finding a potential function \(\Phi(x, y)\).

In the exercise, we checked for exactness but found that the equation wasn't exact. This means alternative methods or conditions might be needed to solve it, like finding an integrating factor that can turn it into an exact equation. Recognizing exact differential equations is important for identifying the simplest resolution path.

Calculus

Calculus is the branch of mathematics examining how things change. It provides the tools necessary for modeling real-world phenomena and solving problems effectively.

In the context of differential equations, calculus provides methods to find rates of change and to understand the behavior of complex systems through derivatives and integrals.

• Understanding partial derivatives helps grasp how multivariable functions evolve.
• Exactness conditions offer insight into when a derivative can be "reversed" into a function.

Using calculus in differential equations allows for sophisticated analyses of dynamic systems, revealing insights that static equations simply cannot provide. In our example, calculus underpins the methods we use to determine whether an equation is exact, and how it might be solved if it isn't.