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Chapter 8: Q53E (page 356)

Verify that the piecewise-defined function \(y = \left\{ {\begin{array}{*{20}{r}}{ - {x^2},}&{x < 0}\\{{x^2},}&{x \ge 0}\end{array}} \right.\) is a solution of the differential equation \(xy' - 2y = 0\) on \(( - \infty ,\infty )\).

Short Answer

Expert verified

The piecewise-defined function is a solution of the differential equation.

Step by step solution

01

Determine the limits of the function \(y =  - {x^2}\).

Check the continuity of the function at.

The first derivative of the function is,

\(y' = - 2x\)

Substitute \(y\) and \(y'\) in the differential equation.

\(\begin{aligned} x( - 2x) - 2( - {x^2}) &= 0\\0 &= 0\end{aligned}\)

Hence, the left-hand limit is equal to the right-hand limit, so the function is continuous at and the solution is verified.

02

Determine the limits of the function \(y = {x^2}\).

Check the continuity of the function at\(x \ge 0\).

The first derivative of the function is,

\(y' = 2x\)

Substitute \(y\) and \(y'\) in the differential equation.

\(\begin{aligned} x(2x) - 2({x^2}) &= 0\\0 &= 0\end{aligned}\)

Hence, the left-hand limit is equal to the right-hand limit, so the function is continuous at \(x \ge 0\) and the solution is verified.

Hence, the piecewise-defined function is a solution of the differential equation on the interval \(( - \infty ,\infty )\).

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Most popular questions from this chapter

A regular type of laminate is currently being used by a manufacturer of circuit boards. A special laminate has been developed to reduce warpage. The regular laminate will be used on one sample of specimens and the special

laminate on another sample, and the amount of warpage will then be determined for each specimen. The manufacturer will then switch to the special laminate only if it can be demonstrated that the true average amount of warpage for that laminate is less than for the regular laminate. State the relevant hypotheses, and describe the type I and type II errors in the context of this situation.

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)?

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

For the following pairs of assertions, indicate which with our rules for setting up hypotheses and why (the subscripts 1 and 2 differentiate between quantities for two different populations or samples):

a. H0: 碌= 100, Ha: 碌 > 100

b.H0: 蟽= 20, Ha: \(\sigma \le 20\)

c.H0: p鈮� .25, Ha: p= .25

d.H0: 碌1 - 碌2 = 25, Ha: 碌1 - 碌2 > 100

e.H0: \(S_1^2 = S_2^2\) , Ha: \(S_1^2 \ne S_2^2\)

f.H0: 碌= 120, Ha: 碌= 150

g.H0: 蟽1,/蟽2 =1,Ha: 蟽1,/ 蟽2 鈮�1

h.H0p1 鈥� p2 = -.1, Ha: p1 鈥� p2 < -.1

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.
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