Chapter 8: Q13P (page 398)

In Problems 13 to 15, find a solution (or solutions) of the differential equation not obtainable by specializing the constant in your solution of the original problem. Hint: See Example 3.
13. Problem 2

Short Answer

Expert verified

Answer

$y = 1, y = - 1, x = 1$ , and $x = - 1$

Step by step solution

Given information

We have given a differential equation $x \sqrt{1 - y^{2}} d x + y \sqrt{1 - x^{2}} d y = 0$ with the boundary condition $y = \frac{1}{2}$ when $x = \frac{1}{2}$ .

Definition of a separable differential equation

Any equation of the form $\frac{d y}{d x} = f (x) g (y)$ is called separable; that is any equation in which dx and terms involving x can be put on one side and dy and terms involving y on the other. For example, $f (x) d x = g (y) d y$ .

Separate the given differential equation

When we separate the variables in the differential equation, we will get

$\frac{y}{\sqrt{1 - y^{2}}} d y = - \frac{x}{\sqrt{1 - x^{2}}} d x$ .

Divide the left-hand side by $\sqrt{1 - y^{2}}$ and the right-hand side by $\sqrt{1 - x^{2}}$

Now, this step is not valid if $y = 1, y = - 1, x = 1$ , and $x = - 1$ , which are solutions for this differential equation that are not obtained by any choice of C.

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In Problems 13 to 15, find a solution (or solutions) of the differential equation not obtainable by specializing the constant in your solution of the original problem. Hint: See Example 3.
13. Problem 2

Short Answer

Step by step solution

Given information

Definition of a separable differential equation

Separate the given differential equation

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In Problems 13 to 15, find a solution (or solutions) of the differential equation not obtainable by specializing the constant in your solution of the original problem. Hint: See Example 3.13. Problem 2

Short Answer

Step by step solution

Given information

Definition of a separable differential equation

Separate the given differential equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Radiation

Wave Optics

Electricity and Magnetism

Waves Physics

Energy Physics

Study anywhere. Anytime. Across all devices.

In Problems 13 to 15, find a solution (or solutions) of the differential equation not obtainable by specializing the constant in your solution of the original problem. Hint: See Example 3.
13. Problem 2