Problem 2 - Applied Calculus

Question

Check that \(y=t^{4}\) is a solution to the differential equation \(t \frac{d y}{d t}=4 y\)

Step-by-Step Solution

Answer

Yes, \(y = t^4\) is a solution to the differential equation.

Step 1: Differentiate the given function

Start by finding the derivative of the given function with respect to \(t\). For \(y = t^4\), the derivative \( \frac{d y}{d t} \) can be found using the power rule: \( \frac{d y}{d t} = 4t^{3} \).

Step 2: Substitute derivatives into the differential equation

Substitute \(y = t^4\) and \(\frac{d y}{d t} = 4t^3\) into the given differential equation \(t \frac{d y}{d t} = 4 y\). This gives: \( t \cdot 4t^3 = 4 \cdot t^4 \).

Step 3: Simplify the equation

Simplify the expression from Step 2. Calculate \(t \cdot 4t^3 = 4t^4\) and recognize that \(4 \cdot t^4 = 4t^4\).

Step 4: Confirm that the equation holds

After simplification, observe that both sides of the equation are equal: \(4t^4 = 4t^4\). This confirms that \(y = t^4\) satisfies the differential equation \(t \frac{d y}{d t} = 4 y\).

Key Concepts

Solutions Verification

To verify if a function is indeed a solution to a differential equation, start by ensuring that all operations and substitutions are executed correctly. This involves confirming that the function, once placed into the differential equation, makes both sides of the equation equal each other. You begin by determining if the expression satisfies all parts of the equation when inserted as indicated.

A typical process for solutions verification includes:

• Substitution: Plug the given function and its derivative into the equation.
• Check Consistency: Simplify both sides of the differential equation to confirm they are identical.

In our example, we inserted the derived function, integrated it within the differential expression, and after simplification, both sides matched, confirming it as a valid solution.

Power Rule Differentiation

Differentiation is a fundamental aspect in calculus, used to find the rate of change of a function with respect to one of its variables. The power rule is a basic differentiation rule applied to functions of the form, where a variable is raised to a constant power.

The power rule states that if you have a term, say, \( t^n \), the derivative \( \frac{d}{dt}(t^n) \) is calculated as \( n \, t^{n-1} \).
For example, let's differentiate \( y = t^4 \):

• Identify the exponent: Here, \( n = 4 \).
• Apply the power rule: The derivative becomes \( 4t^3 \) because \( 4 \cdot t^{4-1} = 4t^3 \).

The application of the power rule in this problem demonstrates how simple mathematical rules can be used effectively to arrive at solutions, prepared for subsequent steps like substitution.

Substitution in Equations

Substitution is a key step in solving differential equations. It involves replacing variables and functions with their actual values or expressions, helping simplify the equation for easier analysis.

In the process of solving the differential equation, we substituted both the original function and its derivative:

• Original function substitution: Replace \( y \) with \( t^4 \).
• Derivative substitution: Replace \( \frac{dy}{dt} \) with \( 4t^3 \).

This method helps the equation reach a form where further simplification can show that both sides of the equation are balanced. By putting the expressions into the equation \( t \frac{d y}{d t} = 4y \), the substition reflects initial condition making the later simplification clearer: \( t \cdot 4t^3 = 4 \cdot t^4 \), showing equality, hence proving the function is a solution.