Question

Graphs of Particular Solutions In Exercises 33 and \(34,\) the general solution of the differential equation is given. Use a graphing utility to graph the particular solutions for the given values of \(C .\) $$ \begin{array}{l}{4 y y^{\prime}-x=0} \\ {4 y^{2}-x^{2}=C} \\ {C=0, C=\pm 1, C=\pm 4}\end{array} $$

Short answer

The particular solutions \(4y^2 - x^2 = C \) with different 'C' values represent different members of the hyperbola family, with each member distinguished by size and orientation. Specifically, for \(C=\pm1\), the hyperbolas open along y and x axis respectively, while for \(C=\pm4\), the hyperbolas open along the y and x-axis respectively but are larger in size.

Step-by-step solution

Understand the General Solution

First, take a moment to understand the given general solution \(4y^2 - x^2 = C\). This equation represents a family of hyperbolas where 'C' is a constant that determines the specific member (particular solution) of this family.

Graph for C=0

Let's start by graphing the solution for \(C=0\). Substituting this value into the general solution, we get \(4y^2 - x^2 = 0\). This is the equation of a pair of straight lines passing through the origin, with slopes that are different and will form the particular solution for \(C=0\).

Graph for C=1 and C=-1

Proceed similarly by plugging in the values \(C=1\) and \(C=-1\) into the general solution. For \(C=1\), the resulting equation \(4y^2 - x^2 = 1\) represents a hyperbola with center at origin and opens along the y-axis. While for \(C=-1\) or \(4y^2 - x^2 = -1\), it represents a hyperbola that opens along the x-axis.

Graph for C=4 and C=-4

Now plug in the values \(C=4\) and \(C=-4\) into the general equation. For \(C=4\), the new equation \(4y^2 - x^2 = 4\) also represents a hyperbola opening upwards and downwards (langs the y-axis), but the vertices are farther from the origin than in previous step. While for \(C=-4\) or \(4y^2 - x^2 = -4\), it still draws a hyperbola that opens along the x-axis but the vertices are farther from the origin compared to that in the above step.

Understanding the Graphs

From the individual graphs for each specific value of 'C', we see that varying 'C' changes the size and the direction of opening of the hyperbolas. Thus, each 'C' just reflects a different member of the family of solutions to the differential equation.

Key concepts

Particular Solution

When dealing with differential equations, a particular solution is a single, specific solution to the equation that not only satisfies the differential equation but also meets certain initial conditions or constraints. In the context of the exercise, the equation given is \(4y^2 - x^2 = C\), and the particular solutions are found by assigning values to the constant \(C\). This turns the general solution into a particular one—like plugging in a key to unlock a unique scenario from the general case.

For example, when \(C=1\), the resulting equation \(4y^2 - x^2 = 1\) describes the location and shape of one particular hyperbola, a curve specifying certain initial conditions. Graphing this equation allows students to visualize this unique solution, setting it apart from the rest of the family described by the general solution.

General Solution

A general solution to a differential equation includes a family of possible solutions, differentiated from each other by one or more arbitrary constants. In our exercise, the general solution \(4y^2 - x^2 = C\) reflects this concept. The 'C' represents an infinite multitude of possibilities, each corresponding to a different curve—a hyperbola in this case.

The significance of understanding the general solution is in recognizing that it provides the broader picture, representing all possible particular solutions. By graphing multiple values for 'C', students can appreciate the diversity within a family of solutions, observing how changing 'C' alters the shape and position of the hyperbolas relative to the basic form of the equation.

Family of Hyperbolas

The phrase family of hyperbolas refers to a group of curves that share a common general equation, where each member of the family is defined by specific values of one or more constants. In the given differential equation \(4y^2 - x^2 = C\), varying 'C' yields different members of this family.

Hyperbolas have two branches and are symmetric about their center, which in the case of this exercise, is at the origin. The variable 'C' affects the distance of the vertices from the center and the orientation of the hyperbola (whether it opens along the x-axis or y-axis). For instance, positive values of 'C' result in hyperbolas that open up and down, while negative values result in hyperbolas that open left and right. This intricate variety in a single family showcases the power of the constant 'C' in generating diverse geometric shapes.

Graphing Utility

A graphing utility is an indispensable tool for visualizing the solutions of differential equations, especially when dealing with a family of hyperbolas. Utilizing a graphing calculator or computer software, students can easily input the general solution and then modify 'C' to observe the resulting particular solutions. The benefit is immediate feedback and visual representation of how each curve fits within the family.

Graphing utilities are particularly beneficial in education, as they help students move beyond abstract equations and truly understand the relationship between the mathematical expression and its geometric representation. By engaging with a graphing utility, learners can experiment with values and visualize the impact on the graph, aiding in a more intuitive grasp of the material.