Question

Show that \(\phi(x)=e^{x}\) is a solution of \(y^{\prime \prime}=y^{\prime}\) on a certain interval \(I\).

Short answer

#Answer# The function \(\phi(x)=e^{x}\) is a solution to the differential equation \(y^{\prime \prime}=y^{\prime}\) on the interval \(I=(-\infty, \infty)\).

Step-by-step solution

Find the First Derivative of \(\phi(x)\)

We have \(\phi(x)=e^{x}\). To find the first derivative, we just differentiate it with respect to \(x\). This gives us: $$\phi^{\prime}(x)=\frac{d}{dx}(e^{x})=e^{x}$$

Find the Second Derivative of \(\phi(x)\)

Now we want to find the second derivative, so we differentiate \(\phi^{\prime}(x)\) with respect to \(x\) once more: $$\phi^{\prime \prime}(x)=\frac{d^{2}}{dx^{2}}(e^{x})=\frac{d}{dx}(e^{x})=e^{x}$$

Substitute the Derivatives into the Differential Equation

We want to check if \(\phi(x)\) satisfies the differential equation \(y^{\prime \prime}=y^{\prime}\), so we substitute the expressions we found for \(\phi^{\prime}(x)\) and \(\phi^{\prime \prime}(x)\) into the equation: $$y^{\prime \prime}=y^{\prime} \implies e^{x} = e^{x}$$

Check for Identity and State the Interval

Since we have demonstrated that \(e^{x} = e^{x}\), the function \(\phi(x)=e^{x}\) satisfies the given differential equation. The function \(e^{x}\) is defined for all \(x \in \mathbb{R}\), so it is a solution on the entire real line, which means the interval \(I\) is \((-\infty, \infty)\).

Key concepts

First Derivative

The first derivative is a crucial concept in calculus as it represents the rate of change or the slope of a function at any given point. In our example, we began with the function \(\phi(x) = e^{x}\). To find the first derivative, \(\phi'(x)\), we apply differentiation rules. The derivative of an exponential function of form \(e^{x}\) is itself, \(e^{x}\). Thus, \(\phi'(x) = \frac{d}{dx}(e^{x}) = e^{x}\). This step shows how the function's growth or decay is consistent across its domain. Understanding the behavior of the first derivative helps us analyze and predict how functions behave. Recognizing that exponential functions like \(e^{x}\) have a constant derivative sets the stage for exploring more complex calculus concepts involving change and motion.

Second Derivative

The second derivative provides insight into the concavity or curvature of a function, offering a deeper understanding of its growth behavior. For \(\phi(x) = e^{x}\), we found the first derivative as \(\phi'(x) = e^{x}\). To get the second derivative, we differentiate again: \(\phi''(x) = \frac{d^2}{dx^2}(e^{x}) = e^{x}\). When the second derivative is positive, like in this example, it indicates the function is concave up, suggesting consistent growth. In practical terms, understanding the second derivative helps in evaluating turning points and optimizing processes across varied contexts, from physics to economics.

Solution Verification

Solution verification in the context of differential equations involves checking whether a given function satisfies the equation. Here, our differential equation is \(y'' = y'\). We obtained \(\phi'(x) = e^{x}\) and \(\phi''(x) = e^{x}\). Plugging these into our equation, we see \(y'' = y' \implies e^{x} = e^{x}\), confirming that the function satisfies the equation. This step is fundamental as it validates the function's behavior over the specified interval. Moreover, it reassures that the function is indeed a solution and helps ensure accuracy in more complex mathematical models. Checking solutions like this helps solidify understanding and ensures that mathematical operations result in logical, valid conclusions.

Exponential Function

The exponential function, represented as \(e^{x}\), is indispensable in various scientific and mathematical fields. Its most important characteristic is that its rate of change is proportional to its current value, making it a natural model for growth and decay processes. Exponential functions are defined and differentiable across all real numbers, providing a smooth model for continuous change. This continuity and consistency explain why \(e^{x}\) is a solution over the entire real line \((-\infty, \infty)\). The simplicity of exponential growth is powerful in modeling situations like population growth, radioactive decay, and compound interest calculations. Understanding exponential functions enhances comprehension of growth dynamics, making them foundational to applied mathematics and beyond.