Question

Find an autonomous differential equation that possesses the specified properties. [Note: There are many possible solutions for each exercise.] A differential equation with no equilibrium solutions and \(y^{\prime}>0\) for all \(y\).

Short answer

Question: Provide an example of an autonomous differential equation that has no equilibrium solutions and has a positive derivative for all values of y. Answer: An example of such a differential equation is \(y'(t) = e^{y(t)}\), as it has no equilibrium solutions and always has a positive derivative for all values of \(y\).

Step-by-step solution

Recall the general form of an autonomous differential equation

An autonomous differential equation has the general form \(y'(t) = f(y(t))\). Our goal is to find a function \(f(y)\) that satisfies the given properties.

No equilibrium solution

An equilibrium solution is a constant function, \(y(t) = c\), such that \(y'(t) = 0\). In this case, it means that \(f(c) = 0\). Since we want no equilibrium solution, we need to find a function \(f(y)\) such that \(f(y) \neq 0\) for all \(y\).

Positive derivative

We are given that \(y'(t) > 0\) for all values of \(y\). Since \(y'(t) = f(y(t))\), this implies that \(f(y) > 0\) for all \(y\).

Find an example that fits the conditions

One simple example of a function that is always positive and never equal to 0 is \(f(y) = e^{y}\). Hence, a possible differential equation that meets the conditions is: $$ y'(t) = e^{y(t)} . $$ This differential equation has no equilibrium solutions and a positive derivative for all values of \(y\).

Key concepts

Equilibrium Solutions in Autonomous Differential Equations

In the context of autonomous differential equations, equilibrium solutions play a critical role. But what exactly is an equilibrium solution? Simply put, it's a constant solution to a differential equation, where the rate of change, or derivative, is zero. This means that if you plug in this constant into the function, the derivative becomes zero indicating no change over time.
However, in cases where we're tasked with finding a differential equation that has no equilibrium solutions, we look for a function where the derivative never equals zero. This implies that the system never stays at rest; it's always changing. An equation having no equilibrium solutions ensures there's no state where the system remains static.
In our exercise, we specifically seek a function where, for any value of \( y \), the derivative is never zero. Such a function guarantees that the autonomous differential equation has no equilibrium solutions, meaning \( f(y) eq 0 \) for all possible values of \( y \).

Understanding Positive Derivatives

A positive derivative refers to the rate of change being above zero. In simple terms, it means that as time progresses, the function's output increases. This property is crucial in dynamic systems involving autonomous differential equations, ensuring the function consistently moves upward.
When seeking an equation with strictly positive derivatives, we're setting a condition where the rate of change is always greater than zero. This results in a continuously increasing function without any stationary points.
An important aspect to consider is that a positive derivative aligns with natural growth processes, where values escalate over time. In mathematical terms, for a function \( y'(t) = f(y) \), setting \( f(y) > 0 \) for all \( y \) ensures that at no point does \( y'(t) \), or the rate of change, fall to zero or decline.

Exploring the Exponential Function

The exponential function \( e^y \) is a mathematical representation of exponential growth. It's a prominent example of a function that does not possess equilibrium solutions when considered in differential equations. Why? Because the exponential function is always positive. It never equals zero, precisely why it's perfect for our solution.
The nature of the exponential function \( e^y \) ensures that its derivative is always positive regardless of the value of \( y \). This characteristic makes it a suitable candidate for forming differential equations without equilibrium solutions and with a positive derivative for all possible \( y \).
Moreover, the exponential function represents natural phenomena like population growth and radioactive decay, where systems grow continuously over time. In our specific case, choosing \( y'(t) = e^{y(t)} \) confirms an ever-increasing function, aligning perfectly with the problem's criteria of more than zero growth at all times.