Question

For each of the following differential equations, separate variables and find a solution containing one arbitrary constant. Then find the value of the constant to give a particular solution satisfying the given boundary condition. Computer plot a slope field and some of the solution curves. \(x \sqrt{1-y^{2}} d x+y \sqrt{1-x^{2}} d y=0, \quad y=\frac{1}{2}\) when \(x=\frac{1}{2}\)

Short answer

The particular solution is: \[ -\sqrt{1 - x^2} = \sqrt{1 - y^2} - \sqrt{3} \].

Step-by-step solution

- Identify and Rewrite the Differential Equation

Start with the given differential equation: oindent\( x \sqrt{1 - y^2} dx + y \sqrt{1 - x^2} dy = 0 \).We need to separate the variables by getting all terms involving \(x\) on one side and all terms involving \(y\) on the other.

- Separate Variables

Rearrange the equation to separate the variables:\( x \sqrt{1 - y^2} dx = - y \sqrt{1 - x^2} dy \).Then, split the variables to each side:\( \frac{x}{\sqrt{1 - x^2}} dx = - \frac{y}{\sqrt{1 - y^2}} dy \).

- Integrate Both Sides

Integrate both sides to find the general solution:\[ \int \frac{x}{\sqrt{1 - x^2}} dx = \int - \frac{y}{\sqrt{1 - y^2}} dy \].Use the substitution \( u = 1 - x^2 \) and \( v = 1 - y^2 \):\[ -\frac{1}{2} \int \frac{du}{\sqrt{u}} = \int - \frac{dy}{\sqrt{1 - y^2}} \].Solving these integrals yields:\[ -\sqrt{1 - x^2} = \sqrt{1 - y^2} + C \],where \( C \) is the integration constant.

- Solve for the Arbitrary Constant

Apply the given boundary condition \( y = \frac{1}{2} \) when \( x = \frac{1}{2} \):Substitute these values into the general solution:\[ -\sqrt{1 - (\frac{1}{2})^2} = \sqrt{1 - (\frac{1}{2})^2} + C \],which simplifies to:\[ -\frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} + C \].Solving for \( C \), we get:\[ C = - \sqrt{3} \].

- Formulate the Particular Solution

Substitute the value of \( C \) back into the general solution:\[ -\sqrt{1 - x^2} = \sqrt{1 - y^2} - \sqrt{3} \].Squaring both sides, we get the particular solution.

- Plot the Slope Field and Solution Curves

Using graphing software or a computational tool, plot the slope field for the differential equation and add the solution curve satisfying the boundary condition (\( x \) from \(-1\) to 1 and \( y \) from -1 to 1).

Key concepts

Differential Equations

A differential equation is an equation that involves an unknown function and its derivatives. These equations play a vital role in various fields such as physics, engineering, and economics. In this exercise, we deal with a differential equation that includes functions of both x and y:
\[ x \, \sqrt{1 - y^2} \, dx + y \, \sqrt{1 - x^2} \, dy = 0 \].
To solve this, we must use different techniques, such as separation of variables, to find a function that satisfies the initial equation. This process often results in a solution containing arbitrary constants that must be determined using boundary conditions.

Boundary Conditions

Boundary conditions are additional constraints provided along with a differential equation. These conditions help us find a particular solution from the general solution by setting specific values for the unknown function at given points.
For instance, in the given exercise, the boundary condition is:
\[ y = \frac{1}{2} \text{ when } x = \frac{1}{2} \].
By substituting these specific values into our general solution, we can solve for the integration constant and get a specific answer that satisfies both the differential equation and the boundary condition.

Integration Techniques

Integration techniques are essential tools for solving differential equations, especially when variables are separated. To integrate successfully, you might need to use different methods, such as substitution:
- Start by rearranging the terms to isolate each variable.
- For our differential equation, after separating the variables, we get:
\[ \frac{x}{\sqrt{1 - x^2}} \, dx = - \frac{y}{\sqrt{1 - y^2}} \, dy \].
- Apply appropriate substitutions to simplify the integrals. For instance, let \( u = 1 - x^2 \) and \( v = 1 - y^2 \).
- The integration then proceeds as:
\[ -\frac{1}{2} \int \frac{du}{\sqrt{u}} = \int - \frac{dy}{\sqrt{1 - y^2}} \].
- Execute the integrals and revert the substitutions to find the general solution.

Slope Fields

Slope fields are graphical representations showing the behavior of solutions to differential equations without solving them analytically. They visualize the direction of the curve at various points on the plane.
To plot a slope field:

• Calculate the slope \( \frac{dy}{dx} \) for different points in the plane.
• Draw small line segments at those points to indicate the slope.

For our exercise, this involves computing the slope field for the differential equation \[ x \sqrt{1 - y^2} dx + y \sqrt{1 - x^2} dy = 0 \]. You can use graphing software or computational tools to plot these fields. This visual aid complements the solution process and helps verify the correctness of the particular solution curves derived using boundary conditions.