Problem 10 - Advanced Engineering Mathematics

Question

In Problems 7-12, match each of the given differential equations with one or more of these solutions: (a) \(y=0\), (b) \(y=2\) (c) \(y=2 x\) (d) \(y=2 x^{2}\). $$ x y^{\prime}=y $$

Step-by-Step Solution

Answer

Solutions are (a) \( y=0 \) and (c) \( y=2x \).

Step 1: Understand the Differential Equation

The given differential equation is \( x y' = y \). This is a first-order linear differential equation. Our task is to check which of the given solutions satisfy this equation.

Step 2: Check Solution Option (a)

For \( y = 0 \), the derivative \( y' = 0 \). Substitute into the differential equation:\[x(0) = 0\]This is true for any \(x\). Hence, \( y = 0 \) is a solution.

Step 3: Check Solution Option (b)

For \( y = 2 \), the derivative \( y' = 0 \). Substitute into the differential equation:\[x(0) = 2\]This does not satisfy the equation, so \( y = 2 \) is not a solution.

Step 4: Check Solution Option (c)

For \( y = 2x \), the derivative \( y' = 2 \). Substitute into the differential equation:\[x(2) = 2x\]Simplifying gives \( 2x = 2x \), which is true. Hence, \( y = 2x \) is a solution.

Step 5: Check Solution Option (d)

For \( y = 2x^2 \), the derivative \( y' = 4x \). Substitute into the differential equation:\[x(4x) = 2x^2\]Simplify to \( 4x^2 = 2x^2 \), which is false. So \( y = 2x^2 \) is not a solution.

Step 6: Conclusion

The solutions that satisfy the differential equation \( x y' = y \) are \( y = 0 \) and \( y = 2x \).

Key Concepts

First-Order Linear Differential Equations

A first-order linear differential equation is an equation involving the first derivative of a function. These types of equations can typically be expressed in the form \( y' + P(x)y = Q(x) \). However, in our exercise, the equation is presented in the form \( x y' = y \).

To solve such equations, you often need to manipulate them into a more recognizable form or find solutions that directly satisfy the original equation. The process generally involves:

• Rewriting the equation in standard form if necessary.
• Identifying known solution types or applying specific methods such as separation of variables.

Understanding this structure is crucial as it aids in identifying or verifying potential solutions, as seen in the exercise given.

Solution Verification

Solution verification involves checking whether a proposed answer actually satisfies the requirements of the given differential equation.

For each proposed solution, the primary steps include:

• Calculating the derivative, \( y' \), of the given function.
• Substituting both \( y \) and \( y' \) back into the differential equation.
• Checking if the left-hand side equals the right-hand side consistently for all \( x \).

In our example, we verified several solutions. For instance, for \( y = 0 \), substitution showed the equation held true. Similarly, for \( y = 2x \), substitution simplified to an identity, confirming this as a legitimate solution. This step is vital in mathematical problem solving as it ensures the correctness and applicability of the answers.

Mathematical Problem Solving

This refers to systematically working through a problem to find its solution. It involves:

• Understanding and analyzing the problem statement.
• Formulating a strategy to tackle the problem.
• Implementing the strategy step-by-step while checking for consistency and correctness.
• Concluding with a clear answer.

In the provided exercise, mathematical problem solving began with understanding the differential equation \( x y' = y \), followed by checking potential solutions logically and methodically. The goal is always to reach a solution that is not only correct, but also applicable to the circumstances outlined in the problem.

Educational Textbook Example

Textbook examples play a significant role in education by illustrating how theoretical concepts apply in practice. They provide a structured pathway to learning, often showcasing step-by-step procedures to reach a solution.

The exercise we examined serves as an excellent textbook example. It demonstrates the application of first-order linear differential equations in a clear, easy-to-follow manner. By offering distinct solution options, this example allows students to engage deeply with the content and understand not just the 'how' but also the 'why' behind each step. It emphasizes:

• Clarifying problem expectation.
• Strategically testing possible solutions.
• Learning by verification, enhancing confidence in solution techniques.

These textbook examples are indispensable tools, especially in complex subjects like differential equations, providing clarity and procedural rigour to learners.