Question

Use the method of isoclines to sketch the approximate integral curves of each of the differential equations. \(y^{\prime}=\frac{x}{y}\).

Short answer

First, rewrite the given differential equation \(y' = \frac{x}{y}\) with a constant slope, \(m\). Solve for x, obtaining \(x = my\). Then, consider different values of m and sketch the isoclines (e.g. x = 0, x = y, x = -y). Along these isoclines, draw tangent lines with the same slope (m) and use them as a guide to sketch the integral curves following the direction of the tangent lines without crossing the isoclines.

Step-by-step solution

Rewrite the Equation with Constant Slope

First, rewrite the given differential equation \(y' = \frac{x}{y}\) by introducing a constant m for slope: \(m = \frac{x}{y}\)

Solve for x

Now, solve the equation from Step 1 for x: \(x = my\)

Consider Various Slopes

Next, consider various values of m and see how the equation from Step 2 changes with each slope. For example, if m = 0, then x = 0, and the isocline is the y-axis; if m = 1, then x = y, and the isocline is the line y = x; if m = -1, then x = -y, and the isocline is the line y = -x.

Sketch the Isoclines

On a coordinate plane, sketch the isoclines found in Step 3 (x = 0, x = y, x = -y) as well as other isoclines for different values of m. Make sure to label each isocline with the corresponding slope (m).

Draw Tangent Lines

Along the isoclines, draw short tangent lines with the same slope (m) as the isocline. This gives a sense of the direction the integral curves will follow.

Sketch the Integral Curves

Now, sketch the integral curves by using the tangent lines as a guide. These integral curves follow the tangent lines along the isoclines and should never cross an isocline. After completing these steps, you should have sketched the approximate integral curves of the given differential equation \(y' = \frac{x}{y}\) using the method of isoclines.

Key concepts

Integral Curves

Integral curves are fundamental to understanding differential equations. They represent solutions to a differential equation as paths in a plane. In simpler terms, if you have a slope field, integral curves are like pathways following the direction of the tiny arrows, which indicate the slopes. These curves illustrate how the solution to the differential equation behaves over different values.

• They show how y changes with x under the given differential equation.
• Integral curves are continuous; they flow smoothly across the slope field.
• They usually don't cross each other for a well-defined differential equation.

Integral curves give a visual account of the solutions, helping us understand the behavior over the entire domain without requiring an exact numerical solution.

Differential Equations

Differential equations are equations that relate a function to its derivatives. They express how variables change relative to each other. For our example, the differential equation is represented as \( y' = \frac{x}{y} \). This means the rate of change of \( y \) concerning \( x \) depends on the ratio of \( x \) to \( y \).Differential equations can be:

• Ordinary – involves functions of a single variable and their derivatives.
• Partial – involves functions of multiple variables and their partial derivatives.

Understanding the structure and form of differential equations is essential because it allows us to set up a slope field and eventually sketch integral curves. They are tools to model various phenomena in physics, biology, economics, and beyond.

Slope Fields

Slope fields, also known as direction fields, are graphical representations of differential equations. They begin with plotting short line segments or arrows at various (x, y) points to describe the slope of integral curves. To create a slope field, follow these steps:

• Calculate the slope using the given differential equation at different points.
• Draw a small line or arrow with this slope at each point.
• Repeat for a grid of points to cover the desired area.

The slope field gives an overview of how solutions to the differential equation behave even before sketching integral curves. It visually indicates the direction that a curve starting at any point will follow, essentially acting as a 'roadmap' for integral curves. This assists in understanding the solution's general pattern, stability, and possible equilibria without solving the equation directly.