Q69SE In Problems $15 - 18$ verify t... [FREE SOLUTION]

91影视

Probability And Statistics For Engineering And Sciences

Jay L. Devore

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 714 pages

ISBN: 9781305251809

Chapter 8: Q69SE (page 358)

In Problems $15 - 18$ verify that the indicated function $y = \phi (x)$ is an explicit solution of the given first-order differential equation. Proceed as in Example $6$, by considering $\phi $ simply as a function and give its domain. Then by considering $\phi $ as a solution of the differential equation, give at least one interval $I$ of definition.
$y' = 2x{y^2};y = 1/(4 - {x^2})$

Short Answer

Expert verified

The indicated function is an explicit solution of the given differential equation and the interval is $I:( - \infty , - 2)or( - 2,2)or(2,\infty )$, and the domain is $D:x \ne \pm 2$.

Step by step solution

Define an explicit function.

Anexplicit solutionis one in which the dependent variable is expresseddirectlyin terms of the independent variable and constants.

Let the given function be $y = \frac{1}{{4 - {x^2}}}$.

Then, the first derivative of the function is,

$y' = \frac{{2x}}{{{{(4 - {x^2})}^2}}}$

Determine the explicit solution and its interval.

Substitute $y$ and $y'$ into the left-hand side of the differential equation.

$\begin{aligned}{c}\frac{{2x}}{{{{(4 - {x^2})}^2}}} &= 2x{\left( {\frac{1}{{(4 - {x^2})}}} \right)^2}\\\frac{{2x}}{{{{(4 - {x^2})}^2}}} &= \frac{{2x}}{{{{(4 - {x^2})}^2}}}\end{aligned}$

That is same as the right-hand side of the differential equation. The indicated function is an explicit solution of the given differential equation. Hence the domain of the solution while considering the solution as a function is $D:x \ne \pm 2$. And the interval is $I:( - \infty , - 2)or( - 2,2)or(2,\infty )$.