Problem 11 - Calculus: Early Transcendentals

Question

Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point. \( y' = y - 2x, (1,0) \)

Step-by-Step Solution

Answer

Sketch the direction field first, then draw the solution curve through (1, 0).

Step 1: Understand the Differential Equation

The differential equation given is \( y' = y - 2x \). This is a first-order, linear differential equation where \( y' \) denotes the derivative of \( y \) with respect to \( x \). The expression \( y - 2x \) defines the slope of the direction field at any point \((x, y)\).

Step 2: Create a Grid of Points

To sketch the direction field, choose a grid of points on the \((x, y)\)-plane. Typically, you can select points such as \((x, y)\) where \( x \) and \( y \) range from -2 to 2 with an increment of 0.5, including the point (1, 0) through which the solution curve must pass.

Step 3: Calculate the Slope at Each Point

For each grid point \((x, y)\), compute the slope using the differential equation: \( y' = y - 2x \). Substitute each \( x \) and \( y \) value from your grid into this formula to get the slope that corresponds to each point.

Step 4: Sketch Direction Field Lines

At each point on your grid, draw a small line segment that has the slope calculated in Step 3. These lines represent the direction of the slope field and help visualize how solution curves are expected to behave.

Step 5: Plot the Solution Curve Passing Through (1,0)

Starting at the point (1, 0), follow the slopes indicated by your direction field to sketch the solution curve. This curve represents a particular solution to the differential equation and shows how \( y \) changes with \( x \) starting from that initial point.

Key Concepts

Understanding Differential Equations

Differential equations are types of equations that relate a function with its derivatives. In simple terms, they describe the rate at which something changes. Here, you have a differential equation given as \( y' = y - 2x \). This is interpreted as the rate of change of \( y \) with respect to \( x \). Every solution to this equation represents a function that satisfies this rate of change at every point \((x, y)\). This equation is first-order because it involves only the first derivative of \( y \), which is \( y' \).

What is a Solution Curve?

A solution curve is a graph of a function that satisfies the differential equation throughout its domain. When you sketch a solution curve, you're basically drawing the path that the solution of the equation follows through the coordinate system. In the context of our example \( y' = y - 2x \), the solution curve is the trajectory which corresponds to following the small line segments, starting at the initial condition \((1,0)\). This curve visually shows how \( y \) changes as \( x \) changes, adhering to the rules set by the differential equation.

Exploring Slope Fields

Slope fields, also known as direction fields, provide a visual aid for understanding differential equations. They consist of many short line segments or arrows on a graph, which indicate the slope of the solution curve at various points. These are crucial for visualizing solutions without having to solve the equation analytically. For the equation \( y' = y - 2x \), each point \((x, y)\) on the graph corresponds to a specific slope, given by the expression \( y - 2x \). By plotting these slopes, you create a field that guides the expected direction of any possible solution curve.

First-Order Linear Differential Equations

These are a special class of differential equations that have the form \( y' + p(x)y = q(x) \). They are linear in terms of both the function \( y \) and its derivative \( y' \). In our exercise, it can be rewritten to fit the standard linear form as \( y' - y = -2x \). They are simpler to solve compared to higher-order differential equations and often involve finding an integrating factor. This will then aid in getting an explicit solution or directly sketching solution curves via methods like direction fields. Understanding first-order linear differential equations is fundamental in many fields of science and engineering, where they model a variety of systems.