Question

Create a direction field for the differential equation \(y^{\prime}=x^{2}-y^{2}\) and sketch a solution curve passing through the point \((-1,2)\).

Short answer

Sketch the direction field for \(y' = x^2 - y^2\) and draw a curve through \((-1, 2)\) following the field's slopes.

Step-by-step solution

Understanding the Differential Equation

We are given the differential equation \(y' = x^2 - y^2\). This means that the slope of any solution curve in the \(xy\)-plane at any point \((x, y)\) is given by the expression \(x^2 - y^2\). Our task is to create a direction field for this equation.

Plotting the Direction Field

For various points in the \(xy\)-plane, we calculate \(y' = x^2 - y^2\) to determine the slope at those points. For instance, at point \((0, 0)\), \(y' = 0^2 - 0^2 = 0\). At point \((1, 0)\), the slope is \(1^2 - 0^2 = 1\). At point \((0, 1)\), the slope is \(0^2 - 1^2 = -1\). These calculations can then be used to draw small line segments indicating the slope.

Sketching the Direction Field

Using the calculated slopes, we draw small line segments at various grid points in the \(xy\)-plane. Each segment's slope corresponds to the value of \(y' = x^2 - y^2\) at that point. These segments show the general direction that solution curves will take.

Identifying Initial Conditions

We are asked to find a solution curve that passes through the point \((-1, 2)\). Therefore, we start at the point \((-1, 2)\) on the direction field.

Sketching the Solution Curve

Starting at the point \((-1, 2)\), use the direction field to sketch the solution curve. The segments in the direction field will guide us by indicating the direction and slope of the curve at each point. Connect the points smoothly, following the field's guidance. Ensure the curve fits naturally within the direction field.

Key concepts

Slope Field

A slope field, also known as a direction field, is a graphical representation that helps visualize the solutions of a differential equation. When we deal with a first-order differential equation like \( y' = x^2 - y^2 \), the slope field is created by plotting tiny line segments over a grid of points in the xy-plane. Each segment represents the slope given by the differential equation at that specific point.Here’s a simple way to think about it:

• For each point \((x, y)\), calculate the slope using \( y' = x^2 - y^2 \).
• Draw a small line segment at that point, oriented with the calculated slope.
• Repeat this over a wide range of points to form the slope field.

Slope fields provide a visual overview of how solutions to the differential equation behave. They serve as a guide to predicting the paths and behaviors of solution curves, which we’ll dive into next.

Solution Curve

In the context of a slope field, a solution curve represents a path that adheres to the prescribed slopes at each point. This curve graphically represents a solution to the differential equation \( y' = x^2 - y^2 \). When drawing a solution curve, the slope field acts like a compass, directing how the curve should incline or decline throughout its path.To sketch a solution curve:

• Begin at your given initial point, as specified by the problem.
• Trace the curve by following the direction indicated by the slope segments in the field.
• As you interpret the slopes, ensure the line remains smooth and consistent with the segments.

Solution curves help visualize specific solutions from infinitely many possibilities outlined by the slope field. They are immensely useful in understanding the behavior of solutions to a differential equation under given conditions.

Initial Condition

Initial conditions provide a specific starting point for sketching a solution curve in a slope field. They determine which among the numerous potential solution paths we follow, as each path corresponds to different initial points. In the exercise, the initial condition is given as the point \((-1, 2)\).Here's how initial conditions are used:

• They anchor the solution curve at a specific point in the slope field.
• Once you identify this point, you begin sketching the solution curve from there.
• The initial condition applies to the particular solution of the differential equation, as opposed to the general family of solutions it represents.

By using initial conditions, we pinpoint a unique solution that meets specific criteria, leading us to a clearer understanding of practical applications where set beginnings are predetermined, such as physics or engineering scenarios.