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Exponential functions are mathematical functions of the form \( y = a e^{bx} \), where \( e \) is the base of the natural logarithms, approximately equal to 2.71828. They are widely used in modeling real-world scenarios where change occurs continuously and at a rate proportional to its current value.
Key Characteristics of Exponential Functions:

• They have a constant base \( e \) and a variable exponent.
• The function's growth or decay is determined by the sign and value of the exponent coefficient.
• If \( b > 0 \), the function represents exponential growth, and if \( b < 0 \), it indicates exponential decay.

In the exercise, the function \( y = 5 + 3 e^{kx} \) includes an exponential component \( 3e^{kx} \), where \( k \) dictates growth or decay. Understanding this structure is crucial for integrating and differentiating these functions within differential equations.

Differentiation is a fundamental concept in calculus focusing on finding the derivative of a function. It measures the rate at which a quantity changes as one of its variables changes. When differentiating an exponential function like \( y = 3e^{kx} \), it's vital to apply the chain rule properly. The derivative of \( e^{u} \) with respect to \( x \) is \( e^{u} \cdot \frac{du}{dx} \). This means for a function \( 3e^{kx} \), the derivative \( \frac{dy}{dx} \) becomes \( 3ke^{kx} \).
Key Steps in Differentiating Exponential Functions:

• Identify the inner function and differentiate it.
• Multiply the outer derivative by the derivative of the inner function.
• Simplify the result, if necessary.

In our exercise, differentiating \( y = 5 + 3e^{kx} \) involves deriving only the exponential part since the derivative of a constant (5) is zero. This mastery of differentiation is critical for solving differential equations.

Solving equations, particularly differential equations, involves finding an unknown value or function that satisfies a given equation. In the problem, the task was to find the value of \( k \) such that the expression \( y = 5 + 3 e^{kx} \) satisfies the differential equation \( \frac{d y}{d x} = 10 - 2 y \).Steps to Solve the Equation:

• Differentiate the function to obtain \( \frac{dy}{dx} \).
• Substitute \( y \) and \( \frac{dy}{dx} \) into the differential equation.
• Equate and simplify both sides to find expressions in terms of \( e^{kx} \).
• If \( e^{kx} eq 0 \), divide by \( e^{kx} \) to isolate terms involving \( k \).
• Solve for \( k \) to determine the required values.

In this exercise, after differentiating and substituting back into the equation, the steps revealed that \( k = -2 \) satisfies the original differential equation. Evaluating and solving such equations is essential in fields like physics and engineering, where they describe dynamic systems.