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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

01

Given information

We have given a differential equation x1-y2dx+y1-x2dy=0with the boundary condition y=12whenx=12.

02

Definition of a separable differential equation

Any equation of the form dydx=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)dx=g(y)dy.

03

Separate the given differential equation

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

y1-y2dy=-x1-x2dx.

Divide the left-hand side by 1-y2and the right-hand side by 1-x2

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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