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 \(C .\) $$ \begin{array}{l}{y y^{\prime}+x=0} \\ {x^{2}+y^{2}=C} \\ {C=0, C=1, C=4}\end{array} $$

Short answer

The graphs for the given differential equations representing \( C=0, C=1, C=4 \) are a single point at the origin, a circle with radius of 1, and a circle with radius of 2, respectively.

Step-by-step solution

Understanding the exercise

The implication of the component \( x^{2}+y^{2}=C \) is that for every set of coordinates \( (x,y) \), their squared sum will always be equal to \( C \). This represents a circle with radius \( \sqrt{C} \).

Draw the Graph for C=0

When \( C=0 \), \( x^{2}+y^{2}=0 \) implies either \( x=0 \) and/or \( y=0 \). Therefore, graphing it will give a point at the origin (0,0).

Draw the graph for C=1

When \( C=1 \), it means our circle has a radius \( \sqrt{1} = 1 \). The graph for this equation is a circle centered at the origin with radius 1.

Draw the Graph for C=4

Setting \( C=4 \) implies a radius of \( \sqrt{4}=2 \). Thus the graph for this equation will be a circle centered on the origin with radius 2.

Key concepts

Particular Solutions

In differential equations, a *particular solution* is a solution derived from the general solution by assigning specific values to the arbitrary constants. For a differential equation, the general solution will typically include constants that can be adjusted to fit specific initial conditions. By selecting certain values for these constants, you can find unique functions known as particular solutions that satisfy both the differential equation and any given initial conditions.

For example, in the exercise, the function \(x^2 + y^2 = C\) represents a family of circles centered at the origin. Each particular solution corresponds to a unique circle determined by the value of \(C\). This shows how constants can define the specifics of a geometric shape in the plane.

Graphing Utilities

Graphing utilities are powerful tools that aid in visualizing mathematical equations and their solutions. These tools can range from online graphing calculators to software programs like Desmos or GeoGebra. When working with differential equations, graphing utilities help generate the shapes or curves described by particular or general solutions, making it easier to grasp the relationship between variables.

With graphing utilities, you can input the equation \(x^2 + y^2 = C\) and visualize how the graph changes for different values of \(C\). For instance, as demonstrated in the exercise, when \(C = 1\), the utility will graph a circle of radius 1 around the origin. By adjusting the value of \(C\), you can immediately see the effect on the radius of the circle.

General Solution

The *general solution* of a differential equation includes all possible solutions that can be derived by varying arbitrary constants. It represents the complete set of solutions without any specific initial conditions applied. In mathematical terms, it includes an expression or function that solves the differential equation for any arbitrary set of conditions.

In the given exercise, the equation \(x^2 + y^2 = C\) represents the general solution for a set of circles of any radius \(\sqrt{C}\) centered at the origin. This general solution expresses the maximum information of the system, and by inserting specific values for \(C\), we obtain the particular solutions described earlier.

Implicit Equations

In mathematics, an *implicit equation* involves relationships between the variables without explicitly solving for one variable in terms of others. These equations are especially prevalent in representing curves or geometric shapes, such as circles or ellipses, which are often not easily expressed in a single explicit form.

The equation \(x^2 + y^2 = C\) is a prime example of an implicit equation. Here, neither \(x\) nor \(y\) is isolated; instead, their relationship defines a circular shape. Solving implicit equations might require algebraic manipulation or numerical methods, especially when graphing or calculating specific solutions. Graphing utilities are particularly useful in these cases, assisting in visualizing the entire curve rather than isolating a particular variable.