Question

Slope Field In Exercises \(67-72,\) use a computer algebra system to (a) graph the slope field for the differential equation and (b) graph the solution satisfying the specified initial condition. $$ \frac{d y}{d x}=0.25 y, \quad y(0)=4 $$

Short answer

The differential equation is \(y'=0.25y\). Solving this, we find the solution to be \(y(x)=4e^{0.25x}\). The slope field and the graph of the solution can be created using a computer algebra system.

Step-by-step solution

Identifying the differential equation

The first step is to identify the differential equation given. In this case, the differential equation is \(\frac{d y}{d x}=0.25 y\). This equation describes the rate of change of y with respect to x, which is proportional to the current value of y.

Solve the differential equation

The second step involves finding the function y(x) that satisfies the given differential equation. This is a first-order linear differential equation and can be solved using the method of separation of variables. Shuffling the equation gives \(\frac{d y}{y} = 0.25 dx\). Integrating both sides gives \(\ln|y| = 0.25x + C\). By exponentiating both sides, we obtain \(y(x)=Ce^{0.25x}\).

Apply initial condition

We apply the initial condition provided, \(y(0)=4\), to find the value for C in \(y(x)=Ce^{0.25x}\). It results in \(4=Ce^{0}\) so \(C=4\). Thus, the solution is \(y(x)=4e^{0.25x}\)

Graph the slope field and solution curve

The final step is to graph the slope field for the given differential equation and the solution curve. The slope field gives a visual representation of the differential equation, and each slope in the field corresponds to the value of the derivative at that point. The slope field should be made with a computer algebra system, such as Desmos or GeoGebra, and the solution curve \(y(x)=4e^{0.25x}\) can be added to the plot.

Key concepts

Slope Fields

A slope field is a visual representation that helps us understand differential equations. It consists of small line segments or arrows that show the slope of the solution curves of a differential equation at various points in the plane.
Each segment aligns with the slope defined by the differential equation at that point.
- **Importance in differential equations:** Slope fields provide a way to visualize solutions without solving the equation analytically.- **Creating a slope field:** Use a tool like Desmos or GeoGebra to plot small lines based on the slope, calculated using the given differential equation, for several points.In our scenario, the equation is \(\frac{d y}{d x} = 0.25y\). For any point \((x, y)\), the slope is computed as \(0.25y\), indicating how steeply the solution curve ascends or descends from that point. Slope fields enhance comprehension by showing potential paths without needing explicit solutions.

Initial Conditions

Initial conditions help us find a specific solution from multiple possibilities of a differential equation. They provide us a starting point, such as a specific \(y\) value at a certain \(x\) value, to determine the absolute solution.
- **Initial value problem:** Often written as \(y(x_0) = y_0\), where \(x_0\) is the initial x-value and \(y_0\) is the corresponding y-value.- **Use in solution curves:** Applying initial conditions fixates the arbitrary constant often found in solutions, ensuring a unique path or curve.In this case, the initial condition given is \(y(0)=4\). By substituting \(x = 0\) into \(y(x) = Ce^{0.25x}\) and setting \(y(0) = 4\), the constant \(C\) is resolved, finding our specific solution, \(y(x) = 4e^{0.25x}\) instead of any general form.

Separation of Variables

Separation of variables is a powerful technique used to solve first-order differential equations. It's particularly effective when the equation can be rearranged so that each variable only appears on one side of the equation.
- **Procedure:** 1. Rearrange the equation so that one side depends only on \(x\) and the other solely on \(y\). 2. Integrate both sides.- **In our example:** Given \(\frac{d y}{d x} = 0.25 y\), we split it into \(\frac{d y}{y} = 0.25 \, dx\).Upon integrating both sides, we obtain \(\ln|y| = 0.25x + C\), where \(C\) is a constant of integration. By solving further, the equation simplifies to \(y(x) = Ce^{0.25x}\). This method empowers us to transform and solve equations that at first glance seem complicated.

Solution Curves

Solution curves are visual representations of the explicit solutions for differential equations on a slope field. They depict specific paths that a solution might take, given any initial conditions.
- **Purpose:** They show how the dependent variable changes with respect to the independent variable over time.- **Graphing solution curves:** Once we solve the differential equation and apply initial conditions, the resulting function can be plotted over the slope field.In our activity, after finding \(y(x) = 4e^{0.25x}\), this specific curve is mapped onto the slope field. It follows the direction given by the field's little line segments, illustrating one pathway from infinitely many that fulfill the initial differential equation. This ensures that the visual matches the analytic solution, confirming consistency and accuracy in understanding the problem.