Question

Use a computer algebra system to graph the slope field for the differential equation and graph the solution satisfying the specified initial condition.$$ \begin{array}{l} \frac{d y}{d x}=\frac{10}{x \sqrt{x^{2}-1}} \\ y(3)=0 \end{array} $$

Short answer

Graph the slope field of the differential equation \(\frac{dy}{dx} = \frac{10}{x\sqrt{x^2-1}}\) using a computer algebra system. Then, add a solution curve that passes through the point (3,0) and is consistent with the slope field. This will give the solution to the differential equation satisfying the initial condition y(3) = 0.

Step-by-step solution

Analyzing the differential equation

Take a look at the provided differential equation, \(\frac{dy}{dx} =\frac{10}{x\sqrt{x^2-1}}\). It’s a first-degree ordinary differential equation. Here, the slope of the function is determined by the fraction \(\frac{10}{x\sqrt{x^2-1}}\). It suggests that the slope of the function changes depending on the value of x. As x increases, the denominator decreases, making the whole fraction increase, and thus the slope as well.

Creating the slope field

Using a computer algebra system, plot the slope field for this differential equation. The slope field provides a graphical view of the slopes of the solution curves at various points, giving insight into the behavior of the solutions. The direction of the arrows in the slope field indicates whether the solutions are increasing or decreasing, and the steepness of the arrows indicates the rate of change. The solution curves satisfying the differential equation will be tangent to these slopes.

Include solution curve satisfying the initial condition

Next, a solution curve that passes through the point (3,0) and is consistent with the slope field will be added. This curve represents the solution to the differential equation that satisfies the initial condition y(3) = 0.

Analyzing the solution curve

By observing the solution curve in relation to the slope field, you can gain further understanding of the behavior of the solutions to the differential equation. The slope field plot provides a quick view of the general shape and behavior of the solutions for different initial conditions.

Key concepts

Slope Field

When studying differential equations, a slope field acts as a visual aid, providing us with a glimpse into the potential solutions. Imagine planting a small stick into the ground that aligns with the slope of a hill at every point. This is essentially what a slope field does for differential equations; it's a plot that shows small line segments (or vectors) which represent the slope of a solution passing through each point.

Creating a slope field helps visualize the variety of possible solutions and how they might look without solving the equation. When we look at the slope field for the given equation \( \frac{dy}{dx} =\frac{10}{x\sqrt{x^2-1}} \), we notice that each line segment represents the slope of the solution curve at that point. The pattern formed by these line segments gives us an idea of how the solution curves behave; they might curve upwards, downwards, or even show chaotic patterns depending on the nature of the equation. This information can be essential when trying to understand complex systems or modeling real-world phenomena.

Ordinary Differential Equation

Differential equations are mathematical equations that relate some function with its derivatives. When the equation involves functions of only one variable and its derivatives, we call it an ordinary differential equation (ODE). The equation from our exercise \( \frac{dy}{dx} =\frac{10}{x\sqrt{x^2-1}} \) is an example of an ODE, specifically, a first-order ODE because it involves only the first derivative of the function y(x).

ODEs are used extensively to describe a wide range of natural phenomena such as population growth, heat conduction, wave propagation, and in our exercise, it might imply a physical process where the rate of change (the derivative) depends on the state of the system (the value of x). Solutions to ODEs can sometimes be obtained analytically through mathematical techniques, or numerically using computational methods.

Initial Condition

In the realm of differential equations, the initial condition is a pivotal piece of information that helps us find a specific solution to an ODE. Think of it like having a time machine and knowing exactly where you started your journey, which is crucial to determining your current position. In our exercise, the initial condition is given as y(3) = 0.

This tells us that our solution curve must pass through the point (3, 0). It essentially anchors our solution to a specific point, making it possible to find one particular solution out of the infinitely many. When we use this initial condition, alongside the slope field, we can graph a unique solution curve that fits both the general trend provided by the slope field and the fixed starting point that the initial condition specifies. It's like drawing a path that passes through a specific gate in a field of many possible pathways.