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Chapter 8: Q64SE (page 357)

In Problems \(11 - 14\) verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution.

\(\frac{{dy}}{{dt}} + 20y = 24; y = \frac{6}{5} - \frac{6}{5}{e^{ - 20t}}\).

Short Answer

Expert verified

The indicated function is an explicit solution of the given differential equation in the interval \(( - \infty ,\infty )\).

Step by step solution

01

Define an explicit function.

An explicit solution is one in which the dependent variable is expressed directly in terms of the independent variable and constants.

Let the given function be \(\frac{{dy}}{{dt}} + 20y = 24\).

Then, the first derivative of the function is \(\frac{{dy}}{{dt}} = 24{e^{ - 20t}}\).

02

Determine it is nonlinear.

Substitute \(y\) and \(\frac{{dy}}{{dt}}\) into the left-hand side of the differential equation.

\(\begin{aligned}{c}24{e^{ - 20t}} + 20\left( {\frac{6}{5} - \frac{6}{5}{e^{ - 20t}}} \right) &= 24{e^{ - 20t}} + 24 - 24{e^{ - 20t}}\\ &= 24\end{aligned}\)

That is same as the right-hand side of the differential equation for every real number \(x\). Thus, the indicated function is an explicit solution of the given differential equation in the interval \(( - \infty ,\infty )\).

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Most popular questions from this chapter

A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at 40 mph under specified conditions is known to be 120 ft. It is proposed that the new design be implemented only if sample data strongly indicates a reduction in true average braking distance for the new design.

a.Define the parameter of interest and state the relevant hypotheses.

b.Suppose braking distance for the new system is normally distributed with 蟽= 10. Let \(\overline X \) denote the sample average braking distance for a random sample of 36 observations. Which values of \(\overline x \) are more contradictory to H0 than 117.2, what is the P-value in this case, and what conclusion is appropriate if 伪 = .10?

c.What is the probability that the new design is not implemented when its true average braking distance is actually 115 ft and the test from part (b) is used?

Let 碌 denote the true average radioactivity level (picocuries per liter). The value 5 pCi/L is considered the dividing line between safe and unsafe water. Would you recommend testing H0: 碌= 5 versus Ha: 碌> 5 or H0: 碌= 5 versus Ha: 碌 < 5? Explain your reasoning. (Hint: Think about the consequences of a type I and type II error for each possibility.)

In Problems 25鈥�28 use (12) to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution.

\(x\frac{{dy}}{{dx}} - 3xy = 1;y = {e^{3x}}\int_1^x {\frac{{{e^{ - 3t}}}}{t}} dt\)

In Problems \(15 - 18\) verify that the indicated functionis 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.

\(y' = 25 + {y^2};y = 5tan5x\)

For each of the following assertions, state whether it is a legitimate statistical hypothesis and why:

\(\begin{array}{l}{\rm{a}}{\rm{. H: \sigma > 100}}\\{\rm{c}}{\rm{. H: s }} \le {\rm{.20}}\\{\rm{e}}{\rm{. H:}}\overline {{\rm{ X}}} {\rm{ - }}\overline {\rm{Y}} {\rm{ = 5}}\end{array}\) \(\begin{array}{l}{\rm{b}}{\rm{. H: }}\widetilde {\rm{x}}{\rm{ = 45}}\\{\rm{d}}{\rm{. H: }}{{\rm{\sigma }}_{\rm{1}}}{\rm{/}}{{\rm{\sigma }}_{\rm{2}}}{\rm{ < 1}}\end{array}\)

\({\rm{f}}{\rm{. H: \lambda }} \le {\rm{.01}}\), where \({\rm{\lambda }}\) is the parameter of an exponential distribution used to model component lifetime

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