Problem 36 - Calculus

Question

Give an example of: A differential equation all of whose solutions are increasing and concave up.

Step-by-Step Solution

Answer

The differential equation is \( y'' = y' \).

Step 1: Understanding the Problem

We need to find a differential equation such that all its solutions (functions) are increasing and concave up. This means that if we have a solution function \( y(x) \), it should satisfy \( y'(x) > 0 \) and \( y''(x) > 0 \) for all \( x \).

Step 2: Choosing an Appropriate General Solution

A function that is both increasing and concave up is something like \( y = e^x \). This function's derivative is always positive, and its second derivative is always positive, so \( y = e^x \) is increasing and concave up.

Step 3: Deriving the Differential Equation

Let's consider the function \( y = e^x + C \), where \( C \) is any constant. The first derivative \( y' = e^x > 0 \) and the second derivative \( y'' = e^x > 0 \) for all \( x \). The corresponding differential equation involving the first derivative is \( y' = y-C \). However, a more precise differential equation involving the second derivative that fits any constant \( C \) is \( y'' = y' \), as this guarantees \( y' > 0 \) for solutions like \( y = e^x + C \).

Step 4: Formulating the Answer

Thus, the differential equation \( y'' = y' \) ensures that all solutions, such as \( y = e^x + C \), where \( C \) is a constant, are increasing (since \( y' = e^x \)) and concave up (since \( y'' = e^x \)).

Key Concepts

Increasing Functions

An increasing function is one in which the value of the function grows as its input increases. To determine if a function is increasing, we observe its first derivative. The first derivative, denoted as \( y'(x) \), describes the rate of change of the function. If \( y'(x) > 0 \) for all \( x \) in its domain, the function is increasing.

In the context of our exercise, consider the function \( y = e^x + C \). Its first derivative is \( y'(x) = e^x \), which is always positive because the exponential function \( e^x \) is always greater than zero for real numbers. This ensures that \( y(x) \) grows as \( x \) increases, validating that it is an increasing function.

Therefore, when you encounter a differential equation that maintains \( y'(x) > 0 \), you are observing a solution that represents an increasing function.

Concave Up Functions

The term "concave up" refers to the curvature of a function. When a function is concave up, its graph resembles the shape of a cup, curving upwards. The second derivative, noted as \( y''(x) \), indicates this curvature. When \( y''(x) > 0 \), the function is accelerating upwards, confirming it is concave up.

In our exercise, the function \( y = e^x + C \) also serves as an example of a concave up function. Its second derivative is \( y''(x) = e^x \), which, like the first derivative, is always positive. This constant positivity intimates that the graph of the function continually curves upwards.

Thus, a solution to a differential equation like \( y'' = y' \) results in functions that are unmistakably concave up. This helps us identify the behavior of potential solutions without needing to graph them first.

Derivatives

Derivatives are fundamental in understanding how functions change. The first derivative, \( y'(x) \), gives us the rate of change of the function or how steeply it inclines or declines as \( x \) changes. A positive first derivative means the function is increasing, while a negative one indicates it’s decreasing.

The second derivative, \( y''(x) \), is crucial for determining the curvature of the function’s graph. When \( y''(x) > 0 \), the function is concave up, and when \( y''(x) < 0 \), it’s concave down. These insights are vital in understanding and predicting the behavior of solutions to differential equations.

For example, in the differential equation \( y'' = y' \), both the first and second derivatives being greater than zero ensure that every solution function behaves in a predictable increasing and concave up manner. Recognizing such patterns with derivatives simplifies the analysis of complex mathematical problems.