Question

Use the window \([-2,2] \times[-2,2]\) to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. \(A\) detailed direction field is not needed. $$y^{\prime}(t)=4-y, y(0)=-1$$

Short answer

Question: Sketch the direction field for the differential equation \(y^{\prime}(t)=4-y\) on the interval \([-2, 2] \times [-2, 2]\) and sketch the solution curve that corresponds to the initial condition \(y(0)=-1\). Answer: The direction field is created by calculating the slopes based on the given differential equation, \(y^{\prime}(t)=4-y\). The solution for this differential equation is \(y(t)=4-(4-y_0)e^{-t}\), and using the initial condition \(y(0)=-1\), we find \(y_0=1\). Therefore, the particular solution is \(y(t)=4-3e^{-t}\). By plotting the direction field and overlaying the solution curve, we can observe that the curve converges to \(4\) as \(t\) increases.

Step-by-step solution

Sketch the direction field for the given differential equation and interval.

A direction field gives a visual representation of the trend of the function \(y(t)\). For each point \((t, y)\) within the interval, we can calculate the slope of the tangent line representing the rate of change at that point. The given equation is \(y^{\prime}(t)=4-y\), which represents this slope. We observe that when \(y = 4\), the slope is zero, meaning horizontal lines are expected at this point; and when \(y < 4\), the slope is positive, meaning the lines slope upwards, and when \(y > 4\), the slope is negative meaning the lines slope downwards. The interval given is \([-2, 2] \times [-2, 2]\). So, for example, we may choose some equally spaced points within this interval to represent the direction field. Noting the slope at those points based on the function \(y^{\prime}(t)=4-y\) will give us a rough idea of the direction field.

Solve the given differential equation.

To sketch the curve corresponding to the initial condition, we first need to solve the given differential equation: $$y^{\prime}(t)=4-y$$ We can rewrite this equation as a separable equation: $$\frac{dy}{dt}+y=4$$ $$\int \frac{dy}{(4-y)} = \int dt$$ Applying the integration, we get: $$\int \frac{dy}{(4-y)} = \int dt$$ $$-\ln|4-y|=t+C$$ Rearranging, we get: $$y(t)=4-(4-y_0)e^{-t}$$

Find the constant of integration (C) using the initial condition.

To find the value of \(y_0\), we use the initial condition given, \(y(0)=-1\): $$-1=4-(4-y_0)e^{0}$$ $$-1=4-(4-y_0)$$ $$y_0=1$$ Now, the particular solution corresponding to the initial condition is: $$y(t)=4-(4-1)e^{-t}=4-3e^{-t}$$

Sketch the solution curve on the direction field plot.

Now that we have the particular solution, we can sketch the curve on the direction field plot. Starting from the initial condition \((0,-1)\), we follow the direction of the tangent lines while keeping the shape of the curve in accordance with the particular solution \(y(t)=4-3e^{-t}\). This curve converges to \(4\) as \(t\) increases, which is consistent with the given differential equation (\(y^{\prime}(t)=4-y\)).

Key concepts

Differential Equations

Differential equations are equations that involve the derivatives of a function. These equations describe how a quantity changes with respect to another variable. In our exercise, the differential equation given is \(y'(t) = 4-y\). The piece \(y'(t)\) represents the rate of change of the function \(y(t)\) with respect to time \(t\). Essentially, it's a way of predicting future behavior of \(y(t)\) from its current state.
The core idea behind solving differential equations is finding a function \(y(t)\) that satisfies the given equation. By understanding how this quantity changes, we can model various real-world phenomena like population growth, heat conduction, and more.

Initial Condition

An initial condition is a specific value that an unknown function takes at a certain point. It is crucial for solving differential equations uniquely. In the context of our problem, the initial condition provided is \(y(0) = -1\). It tells us the value of the function \(y\), at the starting point \(t = 0\).
This initial value is utilized after finding the general solution of the differential equation. We substitute this condition into the general solution to solve for any constants and determine the exact, particular solution for our specific scenario. Providing an initial condition helps in narrowing down the infinite set of potential solutions to a single, precise one.

Separable Equations

Separable equations are a specific type of differential equations that can be rewritten such that all terms involving one variable (like \(y\)) are on one side, and all terms involving the other variable (like \(t\)) are on the other side.
In our example, the differential equation \(y'(t) = 4-y\) can be rearranged to isolate variables on each side: \(\frac{dy}{dt} = 4-y\). This can be further rewritten as \(\frac{dy}{4-y} = dt\).
Once separated, these can be integrated on both sides to find the variables in terms of each other: \(\int \frac{dy}{4-y} = \int dt\). This process is essential as it leads to finding the function \(y(t)\). Separable equations make the solving process more straightforward and manageable.

Slope of Tangent Line

The slope of the tangent line provides a graphical representation of a differential equation's solutions at various points. It's literally the "direction" the solution curve takes at a particular point. In our exercise, we work with \(y'(t) = 4-y\), where \(y'(t)\) defines the slope of the tangent line at each point along the function \(y(t)\).
For instance, at points where \(y = 4\), the slope is zero, and the tangent line is horizontal. When \(y < 4\), the slope is positive, indicating the function increases and the line slopes upwards. Conversely, when \(y > 4\), the slope is negative, and the line slopes downwards.

• Understanding these slopes helps in sketching the direction field.
• This visual interpretation allows us to predict how the function will behave.

Direction fields showcase these tangent slopes across different points, guiding the sketching of the solution curve.