Problem 2 - Calculus

Question

Determine whether each function is a solution to the differential equation and justify your answer: $$x \frac{d y}{d x}=4 y.$$ (a) \(y=x^{4}\) (b) \(y=x^{4}+3\) (c) \(y=x^{3}\) (d) \(y=7 x^{4}\)

Step-by-Step Solution

Answer

Solutions: (a) and (d).

Step 1: Understand the Differential Equation

The given differential equation, \(x \frac{d y}{d x} = 4y\), involves both variables \(x\) and \(y\). A solution \(y\) of this equation must satisfy this relationship when its derivative is substituted into the equation.

Step 2: Test Function (a): \(y = x^4\)

First, find the derivative: \( \frac{d y}{d x} = 4x^3 \). Substitute into the differential equation: \(x(4x^3) = 4x^4\). Since \(4x^4 = 4y\), \(y = x^4\) satisfies the equation, confirming it is a solution.

Step 3: Test Function (b): \(y = x^4 + 3\)

Calculate the derivative: \( \frac{d y}{d x} = 4x^3 \). Substitute: \(x(4x^3) = 4x^4\). The right side: \(4(y) = 4(x^4 + 3) = 4x^4 + 12\). Since \(4x^4 eq 4x^4 + 12\), \(y = x^4 + 3\) is not a solution.

Step 4: Test Function (c): \(y = x^3\)

Find the derivative: \( \frac{d y}{d x} = 3x^2 \). Substitute: \(x(3x^2) = 3x^3\). The right side: \(4(y) = 4x^3\). Since \(3x^3 eq 4x^3\), \(y = x^3\) fails to satisfy the equation and is not a solution.

Step 5: Test Function (d): \(y = 7x^4\)

Calculate the derivative: \( \frac{d y}{d x} = 28x^3 \). Substitute: \(x(28x^3) = 28x^4\). The right side: \(4(y) = 4(7x^4) = 28x^4\). Since both sides of the equation match, \(y = 7x^4\) is a solution of the differential equation.

Key Concepts

Derivative Calculation

The process of derivative calculation is crucial in solving differential equations. A derivative represents how a function changes as its input changes.
It is often noted as \( \frac{d y}{d x} \), meaning the rate of change of \( y \) with respect to \( x \).
Here's how to calculate derivatives for our functions:

• Function (a) \( y = x^4 \): The derivative \( \frac{d y}{d x} \) is \( 4x^3 \). This tells us how steep the curve is at any point \( x \).
  
• Function (b) \( y = x^4 + 3 \): Here, the derivative is still \( 4x^3 \). Note that adding a constant does not change the rate of change.
• Function (c): \( y = x^3 \): The derivative is \( 3x^2 \), indicating a different rate of change compared to \( y = x^4 \).
• Function (d): \( y = 7x^4 \): The derivative is \( 28x^3 \), which scales the original rate of change by a factor of 7.

Understanding derivatives helps us predict how a function behaves and confirms if it fits into a differential equation.

Verifying Solutions

In the context of differential equations, verifying a solution means checking if a function satisfies the equation. This involves comparing both sides of the differential equation after substituting the derivative of the function.When testing functions against the differential equation \( x \frac{d y}{d x} = 4y \):

• Function (a) \( y = x^4 \): Substituting the derivative \( 4x^3 \) gives \( 4x^4 \) on both sides, confirming it's a solution.
• Function (b) \( y = x^4 + 3 \): The right side becomes \( 4x^4 + 12 \), which does not match \( 4x^4 \), ruling out this function as a solution.
• Function (c) \( y = x^3 \): Produces \( 3x^3 \) on one side and \( 4x^3 \) on the other, misaligning the equation's requirements.
• Function (d) \( y = 7x^4 \): Both sides equal \( 28x^4 \), verifying it as a solution.

Verifying solutions is crucial for ensuring a function truly fits an equation, ensuring correct modeling of real-world systems.

Substitution Method

The substitution method involves substituting a proposed function and its derivative into a differential equation to determine if it satisfies the equation. This method is straightforward but requires careful derivative calculation.Here's a walkthrough using this method:

• Substitute Derivatives: Begin by calculating the derivative of the function you are testing. For instance, if testing \( y = 7x^4 \), compute its derivative \( 28x^3 \).
• Replace in Differential Equation: Substitute both the derivative and the function into the given equation, \( x \frac{d y}{d x} = 4y \). You get \( x(28x^3) = 4(7x^4) \).
  
• Compare Results: Simplify and compare both sides of the equation. If both are identical, the function is a valid solution. For \( y = 7x^4 \), both sides equate to \( 28x^4 \).

The substitution method can be a powerful tool for testing multiple candidate solutions quickly, allowing efficient verification of results.