Problem 113 - Thomas Calculus

Question

Use a CAS to solve the initial value problems in Exercises \(111-114 .\) Plot the solution curves. $$ y^{\prime}=\frac{1}{\sqrt{4-x^{2}}}, \quad y(0)=2 $$

Step-by-Step Solution

Answer

The solution is \( y = \arcsin\left(\frac{x}{2}\right) + 2 \), defined for \( -2 < x < 2 \).

Step 1: Identify the Differential Equation

The given differential equation is \( y' = \frac{1}{\sqrt{4-x^2}} \) with the initial condition \( y(0) = 2 \). This is a first-order ordinary differential equation (ODE).

Step 2: Solve the Differential Equation Using CAS

Use a CAS (Computer Algebra System) to determine the general solution of the differential equation. The integral of \( \frac{1}{\sqrt{4-x^2}} \) is \( \arcsin(\frac{x}{2}) + C \), where \( C \) is the constant of integration.

Step 3: Apply Initial Condition

Substitute \( x = 0 \) and \( y = 2 \) into the general solution \( y = \arcsin(\frac{x}{2}) + C \) to find the constant \( C \). This gives us: \[ 2 = \arcsin(0) + C \] \[ C = 2 \].

Step 4: Write the Particular Solution

The particular solution to the initial problem is \( y = \arcsin(\frac{x}{2}) + 2 \).

Step 5: Plot the Solution Curve

Use a graphing tool to plot the function \( y = \arcsin(\frac{x}{2}) + 2 \) over an appropriate domain where the solution is defined, typically \( -2 < x < 2 \), to avoid the square root from becoming undefined.

Key Concepts

Initial Value Problem

When dealing with differential equations, we often encounter problems that specify not only the equation itself but also the value of the solution at a specific point. These are known as initial value problems (IVPs). Solving an IVP involves finding a particular solution to a differential equation that satisfies the given initial condition. For instance, in our exercise, we have the differential equation \( y' = \frac{1}{\sqrt{4-x^2}} \) with an initial condition \( y(0) = 2 \). An initial value problem ensures there's a unique solution that not only satisfies the differential equation but also adjusts itself to pass through the specified initial point. The combination of both the differential equation and the initial condition allows us to solve for any constants necessary to define the particular solution.

Computer Algebra System

A Computer Algebra System (CAS) is a powerful software tool designed to perform symbolic mathematics. It can solve equations, integrate functions, and even handle calculus operations. For differential equations, a CAS can automate the process of finding integrals, derivatives, and solutions to complex problems. In our exercise, the CAS was used to solve the differential equation \( y' = \frac{1}{\sqrt{4-x^2}} \). It provided the general solution \( y = \arcsin(\frac{x}{2}) + C \) efficiently and accurately. This is particularly useful in classroom settings and research as it saves time and reduces human error. With CAS, you can quickly explore different scenarios and solutions by just altering the inputs slightly.

Arcsine Function

The arcsine function, denoted as \( \arcsin(x) \), is the inverse of the sine function. It returns an angle whose sine is the given number. By definition, its domain is limited to \(-1 \leq x \leq 1\) and its range is \(-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}\). In mathematics, it serves as a vital component for expressing angles in trigonometric equations. In the context of our solution, the integration of \(\frac{1}{\sqrt{4-x^2}}\) yielded an expression involving \(\arcsin\). Specifically, \( y = \arcsin(\frac{x}{2}) + C \) forms part of the solution, linking the differential equation's integral with trigonometric functions. Through understanding the properties of the arcsine function, we can accurately interpret the behavior of solutions derived from differential equations involving trigonometry.

First-order Differential Equation

A first-order differential equation involves derivatives of a function, with respect to one variable, where the highest derivative is the first. The general form looks like \( y' = f(x, y) \). The equation from our exercise, \( y' = \frac{1}{\sqrt{4-x^2}} \), is an example of such an equation, as it relates the rate of change of \( y \) directly to \( x \). First-order equations can often be solved using methods like separation of variables or integrating factors. In our particular case, an integration approach was utilized. Solving these equations often involves integrating a function, either by hand or using a CAS, which allows us to find the general solution. This general solution is then used, alongside any given initial conditions, to find the specific solution required by the problem.