Chapter 4: Problem 9
Show that the family of circles
$$
\left(x-x_{0}\right)^{2}+y^{2}=1,-\infty
Short Answer
Expert verified
The two first-order differential equations representing the families of upper and lower semicircles are:
1. For the upper semicircles:
$$
\frac{dy}{dx} = \frac{-(x-x_{0})}{\sqrt{1-(x-x_{0})^2}}
$$
2. For the lower semicircles:
$$
\frac{dy}{dx} = \frac{(x-x_{0})}{\sqrt{1-(x-x_{0})^2}}
$$
Step by step solution
01
Differentiate the circle equation with respect to x
Differentiate the given equation with respect to x treating \(y\) as a function of \(x\), but keep in mind that \(x_{0}\) is a constant:
$$
\frac{d}{dx} \left( \left(x-x_{0}\right)^{2}+y^{2} \right) = \frac{d}{dx} (1)
$$
After taking the derivative, we obtain:
$$
2(x-x_{0})+2yy'=0
$$
02
Solve for y' (dy/dx)
Solve the derived equation for y' (dy/dx):
$$
2yy'=-2(x-x_{0})
$$
$$
y'=\frac{-(x-x_{0})}{y}
$$
Now we have the differential equation for the given family of circles:
$$
\frac{dy}{dx} = \frac{-(x-x_{0})}{y}
$$
03
Separating the equations for the semicircles
We have two distinct semicircles for each value of \(x_{0}\). For the first family of semicircles when \(x_{0}<x<x_{0}+1\), we can use the positive square root of the circle equation to represent the upper semicircle:
$$
y = \sqrt{1-(x-x_{0})^2}\quad \text{for} \quad x_{0}<x<x_{0}+1
$$
Similarly, for the second family of semicircles when \(x_{0}-1<x<x_{0}\), we can use the negative square root of the circle equation to represent the lower semicircle:
$$
y = -\sqrt{1-(x-x_{0})^2}\quad \text{for} \quad x_{0}-1<x<x_{0}
$$
Now we have two separate equations for each family of semicircles.
04
Finding the differential equations for the semicircles
For the first family of semicircles, substitute the positive square root of the circle equation into the obtained differential equation, resulting in:
$$
\frac{dy}{dx} = \frac{-(x-x_{0})}{\sqrt{1-(x-x_{0})^2}}
$$
For the second family of semicircles, substitute the negative square root of the circle equation into the obtained differential equation, resulting in:
$$
\frac{dy}{dx} = \frac{-(x-x_{0})}{-\sqrt{1-(x-x_{0})^2}}
$$
Now, we have two differential equations for the families of semicircles:
For the first family (upper semicircles):
$$
\frac{dy}{dx} = \frac{-(x-x_{0})}{\sqrt{1-(x-x_{0})^2}}
$$
For the second family (lower semicircles):
$$
\frac{dy}{dx} = \frac{(x-x_{0})}{\sqrt{1-(x-x_{0})^2}}
$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Differential Equations for Semicircles
Understanding differential equations for semicircles begins with picturing a circle and then focusing on its halves: the upper semicircle and the lower semicircle. To describe these curves mathematically, one can use differential equations, which are a type of equation that relates a function to its derivatives.
Differential equations are powerful tools because they allow us to express rates of change and describe various physical phenomena. When it comes to semicircles, we start with the general equation of a circle and then find a differential equation that the semicircle solutions will satisfy. Specifically, we are interested in the first-order differential equations that describe the slope, or the rate of change of the function y with respect to x (typically written as \(\frac{dy}{dx}\)).
By differentiating the circle's equation with respect to x, we obtained an expression for \(\frac{dy}{dx}\). For the upper and lower semicircles, the only difference in their differential equations is the sign in front of the square root. This reflects the fact that on the upper semicircle, as x increases, y decreases and vice versa for the lower semicircle. It is important to notice the domain restrictions as they ensure that the equation only represents half of the circle. The differential equations we derived represent the gradients of the tangents to these semicircles at any point along their arc.
Differential equations are powerful tools because they allow us to express rates of change and describe various physical phenomena. When it comes to semicircles, we start with the general equation of a circle and then find a differential equation that the semicircle solutions will satisfy. Specifically, we are interested in the first-order differential equations that describe the slope, or the rate of change of the function y with respect to x (typically written as \(\frac{dy}{dx}\)).
By differentiating the circle's equation with respect to x, we obtained an expression for \(\frac{dy}{dx}\). For the upper and lower semicircles, the only difference in their differential equations is the sign in front of the square root. This reflects the fact that on the upper semicircle, as x increases, y decreases and vice versa for the lower semicircle. It is important to notice the domain restrictions as they ensure that the equation only represents half of the circle. The differential equations we derived represent the gradients of the tangents to these semicircles at any point along their arc.
Differentiation with Respect to x
In the realm of calculus, differentiation with respect to x is a fundamental concept. It provides a way to find the rate at which a variable, y, changes as x changes. Think of it as finding the slope of y relative to x at any point on a curve.
To differentiate an equation with respect to x, we apply the rules of differentiation, keeping in mind that any term not involving x is treated as a constant. In the context of our circle equation, we treated \(y\) as a function of \(x\), while \(x_0\), the x-coordinate of the circle’s center, was held constant. This process results in a relation that gives us the derivative of y, represented as \(\frac{dy}{dx}\), which is key to understanding how the circle's curve changes at every point.
The differentiation step in solving our semicircle problem produced a new equation involving the derivative \(\frac{dy}{dx}\). This derivative signifies the semicircle's slope at any given point and is crucial for sketching the direction field or slope field, which gives a visual representation of the solutions to the differential equation.
To differentiate an equation with respect to x, we apply the rules of differentiation, keeping in mind that any term not involving x is treated as a constant. In the context of our circle equation, we treated \(y\) as a function of \(x\), while \(x_0\), the x-coordinate of the circle’s center, was held constant. This process results in a relation that gives us the derivative of y, represented as \(\frac{dy}{dx}\), which is key to understanding how the circle's curve changes at every point.
The differentiation step in solving our semicircle problem produced a new equation involving the derivative \(\frac{dy}{dx}\). This derivative signifies the semicircle's slope at any given point and is crucial for sketching the direction field or slope field, which gives a visual representation of the solutions to the differential equation.
Solving Differential Equations
The process of solving differential equations often involves finding functions that satisfy the equations. Differential equations are ubiquitous in various scientific fields, representing dynamic processes and change.
When we solve a differential equation, we are looking for a function or set of functions (sometimes called the general solution) that can fulfill the equation, meaning when we differentiate this function and substitute it back into the original equation, it holds true for all allowed values of x. In the context of semicircles, finding the differential equation is only part of the task; solving it would mean finding explicit expressions for y in terms of x that satisfy the semicircle's properties for all x within the specified domain.
However, solving differential equations can be complex and does not always result in a simple formula. Some differential equations might require numerical methods for solving, but in our case, the equation is tractable and leads to explicit formulas representing the upper and lower semicircles.
When we solve a differential equation, we are looking for a function or set of functions (sometimes called the general solution) that can fulfill the equation, meaning when we differentiate this function and substitute it back into the original equation, it holds true for all allowed values of x. In the context of semicircles, finding the differential equation is only part of the task; solving it would mean finding explicit expressions for y in terms of x that satisfy the semicircle's properties for all x within the specified domain.
However, solving differential equations can be complex and does not always result in a simple formula. Some differential equations might require numerical methods for solving, but in our case, the equation is tractable and leads to explicit formulas representing the upper and lower semicircles.
Separation of Variables
The technique called separation of variables is among the simplest methods for solving ordinary differential equations, especially those that are first-order. This method puts all terms involving the dependent variable, y, on one side of the equation, and all terms involving the independent variable, x, on the other side.
By cleverly arranging the equation, we can treat \(\frac{dy}{dx}\) almost like a fraction and 'separate' the variables on either side of the equal sign. Once we have done that, the next step is to integrate both sides of the equation, which should yield a function that satisfies the original differential equation. This approach is especially helpful when dealing with physical problems where variables can naturally be separated in this way.
For our semicircle equations, we have already obtained the separated form implicitly by solving for \(\frac{dy}{dx}\). To explicitly solve it, we'd proceed by integrating. The separate forms for the upper and lower semicircles we found allow us to analyze each half of the circle independently, which is particularly useful when considering phenomena occurring solely in one part of the plane, such as water motion in a half-pipe or sound reflection from a parabolic reflector.
By cleverly arranging the equation, we can treat \(\frac{dy}{dx}\) almost like a fraction and 'separate' the variables on either side of the equal sign. Once we have done that, the next step is to integrate both sides of the equation, which should yield a function that satisfies the original differential equation. This approach is especially helpful when dealing with physical problems where variables can naturally be separated in this way.
For our semicircle equations, we have already obtained the separated form implicitly by solving for \(\frac{dy}{dx}\). To explicitly solve it, we'd proceed by integrating. The separate forms for the upper and lower semicircles we found allow us to analyze each half of the circle independently, which is particularly useful when considering phenomena occurring solely in one part of the plane, such as water motion in a half-pipe or sound reflection from a parabolic reflector.
