91Ó°ÊÓ

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.

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 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.