Problem 27 - Advanced Engineering Mathematics

Question

Equating \(M_{y}=3 y^{2}+4 k x y^{3}\) and \(N_{x}=3 y^{2}+40 x y^{3}\) we obtain \(k=10\)

Step-by-Step Solution

Answer

The value of \( k \) is 10.

Step 1: Identify the Functions

We are given two partial derivatives, \( M_y \) and \( N_x \), which are as follows: \( M_y = 3y^2 + 4kxy^3 \) and \( N_x = 3y^2 + 40xy^3 \). These are partial derivatives supposed to come from some functions M and N, which we need to equate with each other.

Step 2: Equate the Given Expressions

The problem asks us to equate \( M_y \) and \( N_x \). Therefore, set the expressions equal to each other: \[ 3y^2 + 4kxy^3 = 3y^2 + 40xy^3. \]

Step 3: Simplify the Equation

Subtract \( 3y^2 \) from both sides of the equation: \[ 4kxy^3 = 40xy^3. \]This simplifies our equation to focus on the terms that include \( xy^3 \).

Step 4: Solve for k

Since the terms \( xy^3 \) are nonzero, they can be canceled out, leading to:\[ 4k = 40. \]Now, solve for \( k \) by dividing both sides of the equation by 4:\[ k = 10. \]

Key Concepts

Functions

In mathematics, functions play a key role in understanding various concepts including partial derivatives. A function is a rule that assigns a unique output for each input from a given set. In this exercise, we deal with two functions denoted by their partial derivatives:

• Partial derivative with respect to y: \( M_y = 3y^2 + 4kxy^3 \)
• Partial derivative with respect to x: \( N_x = 3y^2 + 40xy^3 \)

These derivatives likely originate from larger functions \( M(x, y) \) and \( N(x, y) \), though we do not see these full functions directly in the exercise.
Understanding their role is crucial as partial derivatives measure how a function changes as only one of its variables is varied, keeping others constant. This is useful in multi-variable calculus where decisions are based on several factors. Calculating partial derivatives helps unravel how each factor independently influences the function's output.
A solid grasp of functions and their derivatives allows you to solve more complex equations and systems, as with our problem above.

Equation Simplification

Equation simplification involves reducing expressions to their simplest form to make problem-solving easier. It's like cleaning a messy room to see what's important. In our exercise, we begin with the equation obtained from equating two partial derivatives:
\[ 3y^2 + 4kxy^3 = 3y^2 + 40xy^3 \]Breaking this equation down involves a few key steps:

• Recognize and remove terms common to both sides of the equation. Here, \(3y^2\) is subtracted from both sides.
• This reduces the equation to \[ 4kxy^3 = 40xy^3 \]
• Now, isolate the variable of interest—in this case, \( k \), by canceling like terms. Simplify by dividing through by coefficients or expressions that can be canceled.

By simplifying, we focus on the essential parts of the problem which in this case leads to solving for the constant \( k \). Equation simplification makes equations more manageable and reduces the chances of errors while solving.

Problem Solving

Problem solving with equations, especially involving partial derivatives, requires a strategic and systematic approach. Breaking it down step by step ensures you don't miss vital details along the way. In this exercise, we initially equate two partial derivatives, recognizing them as parts of a system.
The key to solve

• Identify the relationships: Understanding these derivatives come from equations where we equate \( M_y \) to \( N_x \) is crucial.
• Simplify first: By reducing the equation to its simplest form, non-trivial components were canceled naturally, letting us identify things like constants or coefficients cleanly.
• Calculate the unknowns: Finally, solving the simplified equation \( 4k = 40 \) to find \( k = 10 \) answers the problem directly.

This process showcases the importance of logical reasoning, step-by-step simplification, and calculation to find solutions effectively. Remember to stay organized and verify your results to ensure accuracy throughout.