Question

Draw the direction field for the following differential equations, then solve the differential equation. Draw your solution on top of the direction field. Does your solution follow along the arrows on your direction field? $$ \frac{d y}{d x}=x^{2} \cos x $$

Short answer

Yes, the solution should follow the direction of the arrows in the direction field.

Step-by-step solution

Understanding the Differential Equation

The given differential equation is \( \frac{dy}{dx} = x^2 \cos x \). This means the slope of every tangent line in the direction field is determined by the formula \( x^2 \cos x \). The direction field is a graphical representation of these slopes at various points in the plane.

Setting Up the Direction Field

The direction field is constructed by plotting small line segments with slopes given by \( x^2 \cos x \) at various points \((x, y)\). Choose different values of \(x\), substitute them into the equation to find the slope at those points, and draw the corresponding line segments.

Observing the Direction Field

Notice how the slopes change as \(x\) changes. For even values of \(x\) (zero, positive, and negative values), calculate \(x^2 \cos x\) to determine whether the slope is positive, zero, or negative and how steep it is. Observe the pattern to understand the flow of the field.

Solving the Differential Equation

To solve the differential equation \( \frac{dy}{dx} = x^2 \cos x \), integrate both sides. The solution is found by integrating the right side: \[ y(x) = \int x^2 \cos x \, dx + C \] where \(C\) is the constant of integration.

Integration Steps

The integration of \( x^2 \cos x \) can be done using integration by parts multiple times or another technique suited for this integral. Assume the result is some function \( F(x) \) plus the constant \( C \).

Plotting the Solution on the Direction Field

Use the solution \( y(x) = F(x) + C \) and plot this curve on top of the direction field. Check that the curve follows the general direction indicated by the arrows of the direction field.

Verifying Follow Along Arrows

The solution curve \( y = F(x) + C \) should follow along the direction arrows plotted. Check if the curve's tangent matches the slopes given by the direction field at various points.

Key concepts

Direction Field

A direction field is a useful tool for visualizing differential equations. It consists of small line segments or arrows drawn in the plane. Each segment represents the slope of a solution to the differential equation at that point. Here, for the equation \( \frac{dy}{dx} = x^2 \cos x \), the slope is calculated using the expression \( x^2 \cos x \).

To construct a direction field, you begin by choosing a range of \( x \) values. For each \( x \), substitute it into the slope expression to determine the corresponding slope. At each point \((x, y)\), draw a short line segment with the calculated slope. This creates a visual flow that gives an overview of the solution's behavior throughout the plane.

Here are some important tips when working with direction fields:

• Ensure you plot a sufficient number of points to see patterns clearly.
• Notice how slopes change direction or magnitude as \( x \) increases or decreases.
• Identify any symmetry or repeating patterns that can help understand the solution's characteristics.

Integration

Integration is a fundamental process used to solve differential equations, where you find an antiderivative or integral of a given function. For the given problem with \( \frac{dy}{dx} = x^2 \cos x \), solving it involves integrating the right-hand side.

The integral to solve is \( \int x^2 \cos x \, dx \). This can be complex, often requiring techniques like integration by parts. In these situations:

• Start by identifying parts of the function appropriate for these techniques.
• Calculate carefully to avoid mistakes in each step of integration.
• Remember to include the constant of integration, \( C \), in your final solution to account for all possible solutions.

Once integrated, you'll have a function \( y(x) = F(x) + C \), representing the general solution to the differential equation.

Slope Analysis

Analyzing the slopes provided by the differential equation \( \frac{dy}{dx} = x^2 \cos x \) reveals much about the direction field. As \( x \) varies, so does the slope at each point \((x, y)\).

Here's how to intuitively analyze slopes:

• Positive slopes indicate increasin....

Graphical Representation

The graphical representation combines both the direction field and the solution to provide a comprehensive view. First, observe the direction field constructed using slopes \( x^2 \cos x \). These arrows point in the directions that solutions will generally follow.

After integrating to find the solution \( y(x) = F(x) + C \), the next step is plotting this function on the direction field. Observing the plotted solution curve, notice how it aligns with the directional arrows of your field.

Here are some key points to consider:

• Ensure your solution curve follows the direction indicated by the field. If it doesn’t, re-evaluate your calculations or plot points.
• Recognize areas where the solution may diverge or converge with the overall pattern in the direction field.
• Corrections and adjustments to scale and placement might be needed to clearly visualize the relationships.

This integrated view illustrates the dynamic nature of differential equations, making the abstract concept tangible and understandable.