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Chapter 8: Q66SE (page 358)

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.

\(y'' + y = tanx; y = - (cosx) ln(secx + tanx)\).

Short Answer

Expert verified

The indicated function is an explicit solution of the given differential equation.

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.

02

Determine the derivatives of the function.

Let the given function be \(y'' + y = tanx\).

Then, the first derivative of the function is,

\(y' = - (cos x)sec x + sin x ln(sec x + tan x)\)

Using the product rule, the second derivative of the function is,

\(\begin{aligned}{l}y'' &= ( - cos x(sec x tan x) + sin x sec x) + (sin x sec x + cos x ln(sec x + tan x))\\y'' &= - cos x(sec x tan x) + cos x ln( sec x + tan x) + 2sin x sec x\end{aligned}\)

03

Determine the explicit solution and its interval.

Substitute \(y\), \(y'\) and \(y''\) into the left-hand side of the differential equation.

\(\begin{aligned}{c} - cos x(sec x tan x) + cos xln(sec x + tan x) + 2sin x sec x + ( - (cos x)ln(sec x + tan x)) &= tan x\\ - cos x(sec x tan x) + 2sin x sec x &= tan x\\ - \frac{1}{{sec x}}(sec x tan x) + 2sin x\frac{1}{{cos x}} &= tan x\\ - tan x + 2tan x &= tan x\\tan x &= tan x\end{aligned}\)

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.

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Most popular questions from this chapter

The paint used to make lines on roads must reflect enough light to be clearly visible at night. Let \(\mu \) denote the true average reflectometer reading for a new type of paint under consideration. A test of \({H_0}:\mu = 20\) versus \({H_n}:\mu > 20\) will be based on a random sample of size n from a normal population distribution. What conclusion is appropriate in each of the following situations?

\(\begin{array}{l}a.n = 15,t = 3.2,\alpha = .05\\b.n = 9,t = 1.8,\alpha = .01\\c.n = 24,t = - 2\end{array}\)

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.

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

Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40-amp fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40. If the mean amperage is lower than 40, customers will complain because the fuses require replacement too often. If the mean amperage is higher than 40, the manufacturer might be liable for damage to an electrical system due to fuse malfunction. To verify the amperage of the fuses, a sample of fuses is to be selected and inspected. If a hypothesis test were to be performed on the resulting data, what null and alternative hypotheses would be of interest to the manufacturer? Describe type I and type II errors in the context of this problem situation.

A sample of n sludge specimens is selected and the pH of each one is determined. The one-sample t test will then be used to see if there is compelling evidence for concluding that true average pH is less than 7.0. What conclusion is appropriate in each of the following situations?

a.n= 6, t= -2.3, 伪= .05

b.n= 15, t= -3.1伪=.01

c.n= 12, t= -1.3, 伪= .05

d.n= 6, t = .7, 伪 = .05

e.n= 6, \(\overline x = 6.68,s/\sqrt n = .0820\)

Each of a group of \(20\) intermediate tennis players is given two rackets, one having nylon strings and the other synthetic gut strings. After several weeks of playing with the two rackets, each player will be asked to state a preference for one of the two types of strings. Let \(p\) denote the proportion of all such players who would prefer gut to nylon, and let \(X\) be the number of players in the sample who prefer gut. Because gut strings are more expensive, consider the null hypothesis that at most \(50\% \) of all such players prefer gut. We simplify this to \({H_0}:p = .5\), planning to reject \({H_0}\) only if sample evidence strongly favors gut strings.

a. Is a significance level of exactly \(.05\) achievable? If not, what is the largest a smaller than \(.05\) that is achievable?

b. If \(60\% \) of all enthusiasts prefer gut, calculate the probability of a type II error using the significance level from part (a). Repeat if 80% of all enthusiasts prefer gut.

c. If \(13\) out of the \(20\) players prefer gut, should \({H_0}\) be rejected using the significance level of (a)?
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