Problem 50 - Calculus

Question

(a) Sketch the slope field for \(y^{\prime}=-y / x\) (b) Sketch several solution curves. (c) Solve the differential equation analytically.

Step-by-Step Solution

Answer

The differential equation solutions are \( y = \frac{C}{x} \).

Step 1: Understand the Differential Equation

We have the differential equation \( y' = -\frac{y}{x} \). This is a first-order separable differential equation, which we can solve using separation of variables.

Step 2: Sketch the Slope Field

To sketch the slope field, observe that \( y' = -\frac{y}{x} \) gives us the slope for each point \((x, y)\). For each point in the plane, compute \( -\frac{y}{x} \) and draw a small line that represents this slope. For instance, if \( y = x \), then \( y' = -1 \), indicating a slope of -1. Repeat for various points.

Step 3: Sketch the Solution Curves

Solution curves follow the direction given by the slopes in the field. A simple solution is when \( y = 0 \), which means the slope is 0 and the curve is a horizontal line. Other solutions can be sketched considering initial conditions or particular solutions that fit the general behavior shown by the slope field.

Step 4: Separate Variables

Rewrite the equation \( y' = -\frac{y}{x} \) as \( \frac{dy}{y} = -\frac{dx}{x} \). This form allows us to integrate both sides with respect to their respective variables.

Step 5: Integrate Both Sides

Integrate \( \frac{dy}{y} \) to get \( \ln|y| + C_1 \) and integrate \( -\frac{dx}{x} \) to get \( -\ln|x| + C_2 \). Equating the two results, we have \( \ln|y| = -\ln|x| + C \), where \( C = C_2 - C_1 \).

Step 6: Solve for y

To find \( y \), rewrite the equation as \( \ln|y| = -\ln|x| + C \), which implies \( |y| = e^C \cdot \frac{1}{|x|} \). Choosing \( C = \ln(C') \), we get \( y = \frac{C'}{x} \), where \( C' \) is a constant that could be positive or negative.

Key Concepts

Slope Field

A slope field, also known as a direction field, is a visual representation of a differential equation of the form \( y' = f(x, y) \). It consists of small line segments or arrows drawn at various points in the \(xy\)-plane, each one indicating the slope \( y' \) at that point.

To sketch the slope field for the differential equation \( y' = -\frac{y}{x} \), you'll need to compute the slope \( -\frac{y}{x} \) at several points. Here's a quick way to do it:

• Pick several points ((x, y)) in the plane.
• For each point, calculate the slope using the formula \( -\frac{y}{x} \).
• Draw a small line segment representing this slope.

This method provides a field of slopes that gives a quick overview of the solution behavior, helping us to visualize the directional flow of potential solution curves. For instance, when \( y = x \), the slope is -1, indicating that the curves point downwards at a 45-degree angle.

Separation of Variables

Separation of variables is a technique often used to solve first-order differential equations. It involves rearranging the equation so that each variable appears on only one side.

For the equation \( y' = -\frac{y}{x} \), we can rewrite it as \( \frac{dy}{y} = -\frac{dx}{x} \). This is a critical step because it allows us to integrate each side separately. Here's how the process works:

• Move all terms involving \( y \) to one side and all terms involving \( x \) to the other.
• The form \( \frac{dy}{y} = -\frac{dx}{x} \) indicates that both sides are prepared for integration.

This technique is powerful because it transforms a differential equation into two simpler integral equations.

Solution Curves

Solution curves are paths in the \(xy\)-plane that satisfy the differential equation. Once the slope field is sketched, these curves align with the slopes, showing the solution over different initial conditions.

To draw solution curves:

• Identify special or simple solutions, such as \( y = 0 \), where the slope is zero, forming a horizontal line.
• Choose initial conditions or specific values to find particular solution curves.
• Trace curves smoothly so that they follow the direction and steepness indicated by the slope field.

Solution curves paint a complete picture of how variables change over time, offering insights into the behavior of the differential equation across various conditions.

Integration

Integration allows us to find the actual solution from a separable differential equation. By integrating, we can uncover the relationship between the dependent and independent variables.

For the equation \( \frac{dy}{y} = -\frac{dx}{x} \):

• Integrate \( \frac{dy}{y} \) to get \( \ln|y| + C_1 \).
• Integrate \( -\frac{dx}{x} \) to obtain \( -\ln|x| + C_2 \).
• Set these results equal: \( \ln|y| = -\ln|x| + C \), where \( C = C_2 - C_1 \).

This results in \( |y| = e^C \cdot \frac{1}{|x|} \). Choosing \( C = \ln(C') \), the solution becomes \( y = \frac{C'}{x} \), revealing how \( y \) varies with \( x \) given any specific constant \( C' \). In this way, integration transforms a differential equation into its explicit solution.