Question

Find the equilibrium solution of the following equations, make a sketch of the direction field, for \(t \geq 0,\) and determine whether the equilibrium solution is stable. The direction field needs to indicate only whether solutions are increasing or decreasing on either side of the equilibrium solution. $$u^{\prime}(t)+7 u+21=0$$

Short answer

Solution: The equilibrium solution is u(t) = -3, and it is stable.

Step-by-step solution

1. Find the equilibrium solution

In order to find the equilibrium solution, we need to set the derivative equal to zero and solve for the function u(t): $$u^{\prime}(t)+7 u+21=0$$ If \(u^{\prime}(t) = 0\), then: $$7 u(t)+21=0.$$ Now solve for u(t): $$u(t)=-\dfrac{21}{7}=-3.$$ The equilibrium solution is \(u(t)=-3\).

2. Determine the direction of increasing and decreasing solutions

We can now analyze the equation to determine the direction of the solutions, whether they are increasing or decreasing on either side of the equilibrium solution. For this purpose, we will investigate the sign of \(u^{\prime}(t)\) when \(u(t)\) is slightly larger or smaller than the equilibrium solution. If \(u(t) > -3\): $$u^{\prime}(t)+7 u(t)+21 > 0 \Longrightarrow u^{\prime}(t) > 0.$$ This means the solution is increasing on the right side of the equilibrium solution. If \(u(t)<-3\): $$u^{\prime}(t)+7 u(t)+21<0 \Longrightarrow u^{\prime}(t) < 0.$$ This means the solution is decreasing on the left side of the equilibrium solution.

3. Determine if the equilibrium solution is stable

Now that we know the direction of the solutions on both sides of the equilibrium solution, we can determine its stability. A stable equilibrium solution would have solutions moving towards it, while an unstable equilibrium solution would have solutions moving away from it. Here, we found that the solution is increasing when \(u(t) > -3\) and decreasing when \(u(t) < -3\). This means solutions are moving towards the equilibrium solution from both sides. Therefore, the equilibrium solution is stable.

4. Summary

In conclusion, the equilibrium solution of the given linear differential equation is \(u(t) = -3\). Based on our analysis of the direction field, we determined that the solutions on both sides of the equilibrium solution are moving towards it. Hence, the equilibrium solution is stable.

Key concepts

Linear Differential Equation

A linear differential equation is an equation involving the derivatives of a function. In a simple form, it is written as \[ a(t) \frac{du}{dt} + b(t)u = c(t) \] where \( a(t), b(t), \) and \( c(t) \) are functions of the independent variable \( t \).

The equation given in our exercise is \[ u^{\prime}(t) + 7u + 21 = 0. \]

Since the terms involve a linear function of \( u \) and its derivative \( u'(t) \), this qualifies as a first-order linear differential equation. Solving this type of equation often involves finding an equilibrium solution, which is a constant solution where the rate of change of \( u \) is zero.

Here’s what makes a linear differential equation special:

• They can often be solved using straightforward algebraic manipulation.
• The superposition principle applies, meaning any linear combination of solutions is also a solution.
• They are extensively used in modeling real-world phenomena, such as electrical circuits and population growth.

Stability Analysis

Stability analysis involves determining whether an equilibrium solution is stable or not. An equilibrium solution is stable if, when the system is slightly disturbed, it tends to return to the equilibrium point.

For the equation \[ u^{\prime}(t) + 7u + 21 = 0, \] our equilibrium solution was found to be \( u(t) = -3 \).

Here's how we examine stability in this context:

• If, near the equilibrium, the solutions tend to move towards \( u = -3 \), the solution is stable.
• If they tend to move away, it is unstable.
• Neutral stability implies no tendency to move towards or away.

Since the solution was decreasing on one side and increasing on the other, solutions will converge towards \( u = -3 \), indicating a stable solution.

Direction Field

A direction field (or slope field) is a graphical representation that illustrates the slopes of a differential equation without actually solving it.

In the context of our exercise, drawing a direction field for \[ u^{\prime}(t) + 7u + 21 = 0 \] can help visualize how solutions behave around the equilibrium point \( u = -3 \).

Here's what you would typically do:

• Plot the equilibrium point on a graph. For this, it would be the horizontal line at \( u = -3 \).
• Look at the behavior on either side of this point. If above \( u = -3 \), the direction suggests increasing motion; if below, decreasing.
• The direction field gives an intuitive understanding of how solutions change over time.

Direction fields are a great way to visually assess stability by showing if the curve solutions "bend" towards or away from the equilibrium.

Differential Calculus

Differential calculus is a branch of mathematics that deals with rates of change and slopes of curves. It is fundamental in analyzing differential equations.

In our particular context, differential calculus allows us to determine where the function \( u(t) \) does not change – which is how we find equilibrium solutions.

Key concepts include:

• Derivatives: Indicate the rate of change of a function. Here, \( u'(t) \) represents how \( u \) changes over time.
• Solving \( u'(t) + 7u + 21 = 0 \) shows how different values of \( u \) affect its rate of change.
• The technique of "setting the derivative to zero" helps find where functions do not change, pinpointing equilibrium.

Differential calculus empowers you to interpret dynamic systems and understand their behaviors by examining changes and tendencies using derivatives.