Question

In cach of Problems 7 through 10 write down a differential equation of the form \(d y / d t=a y+b\) whose solutions have the required bchavior as \(t \rightarrow \infty\). All solutions approach \(y=2 / 3\)

Short answer

Question: Find the differential equation of the form \(\frac{dy}{dt} = ay + b\) such that as \(t \rightarrow \infty\), all solutions approach \(y = \frac{2}{3}\). Answer: \(\frac{dy}{dt} = 3y - 2\).

Step-by-step solution

Analyzing the required asymptotic behavior

As \(t \rightarrow \infty\), all solutions approach \(y = \frac{2}{3}\). This means that, in the limit, the slope of the function \(y(t)\) becomes zero. Therefore, as \(t \rightarrow \infty\), we must have: $$\lim_{t \to \infty} \frac{dy}{dt} = 0.$$ We can use this information to deduce the required values of \(a\) and \(b\) in the equation \(\frac{dy}{dt} = ay + b\).

Determine the coefficients a and d

Since we know that the limit of \(\frac{dy}{dt}\) goes to zero when \(t \rightarrow \infty\), we can plug the value of \(y = \frac{2}{3}\) into our equation and set the limit to zero: $$\lim_{t \to \infty} \left( a \frac{2}{3} + b \right) = 0.$$ The limit of a sum is equal to the sum of the limits, and since \(a\) and \(b\) are constants, we can take the limit directly: $$a \frac{2}{3} + b = 0.$$ By now, we have a linear equation that relates \(a\) and \(b\): $$2a + 3b = 0.$$ There are infinitely many values of \(a\) and \(b\) that can satisfy this condition. We can choose any one of them, for example, \(a = 3\) and \(b = -2\). So, the differential equation with the required asymptotic behavior is: $$\frac{dy}{dt} = 3y - 2.$$

Key concepts

Asymptotic Behavior

In the study of differential equations, asymptotic behavior describes how solutions behave as time progresses to infinity. When we say that a solution approaches a certain value, it means that as the variable of time, often denoted as \( t \), increases without bound, the solution will come closer and closer to a specific value. In the given problem, all solutions approach \( y = \frac{2}{3} \). This scenario suggests that as time goes on, no matter what the initial starting value of \( y \) was, the function \( y(t) \) will eventually settle or stabilize near the value \( \frac{2}{3} \).

This stabilization means that the rate of change, or slope of \( y(t) \), becomes zero at infinity. Mathematically, this is expressed as:

• \( \lim_{t \to \infty} \frac{dy}{dt} = 0 \)

This condition is pivotal for determining the constants in the differential equation \( \frac{dy}{dt} = ay + b \).

Understanding asymptotic behavior is crucial because it provides insights about the long-term trends of dynamic systems modeled by differential equations.

Linear Differential Equation

A linear differential equation is an equation that involves an unknown function and its derivatives. The presence of the function \( y \) and its first derivative \( \frac{dy}{dt} \) in a linear form, as in the given problem, constitutes a linear differential equation. It can be represented in the standard form:

• \( \frac{dy}{dt} = ay + b \)

Here, \( a \) and \( b \) are constants, and \( y \) is a function of \( t \). The term 'linear' implies that the dependent variable and its derivatives appear to the power of one and are not multiplied by each other.

These equations are fundamental in modeling situations where quantities change at rates that depend linearly on current values. They are widely used in fields like physics, engineering, biology, and economics to model growth, decay, population dynamics, and many other processes.
The linearity of these equations often allows for straightforward solutions and analysis, making them invaluable tools for both theoretical and applied mathematics.

Solution of Differential Equation

A solution to a differential equation is a function, or a set of functions, that satisfies the equation. Finding a solution involves integrating the differential equation and, if needed, applying initial or boundary conditions to determine the constants of integration. In the process outlined in the exercise, the goal was to find values for constants \( a \) and \( b \) such that the solution exhibits the described asymptotic behavior.

In our case, given \( y = \frac{2}{3} \) as the long-term approach value, we set \( \frac{dy}{dt} = ay + b \) to zero at \( y = \frac{2}{3} \) to find:

• \( 2a + 3b = 0 \)

This formula defines a relationship between \( a \) and \( b \). From this equation, we can choose suitable constants such as \( a = 3 \) and \( b = -2 \) which solve the equation.
The resulting differential equation, \( \frac{dy}{dt} = 3y - 2 \), ensures that as \( t \to \infty \), \( y \to \frac{2}{3} \), effectively capturing the required asymptotic behavior. Any specific solution that follows this form will align with the desired long-term trends indicated by the problem.