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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

To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of welds is selected, and tests are conducted on each weld in the sample. Weld strength is measured as the force required to break the weld. Suppose the specifications state that mean strength of welds should exceed 100 lb/in2 ; the inspection team decides to test H0: 碌= 100 versus Ha: 碌> 100. Explain why it might be preferable to use

this Ha rather than 碌 < 100.

Pairs of P-values and significance levels, 伪, are given.

For each pair, state whether the observed P-value would lead to rejection of H0 at the given significance level.

a.P颅-value = .084, 伪= .05

b.P颅-value = .003, 伪= .001

c.P-颅value = .498, 伪= .05

d.P-颅value = .084, 伪= .10

e.P-颅value = .039, 伪= .01

f.P-颅value = .218, 伪 = .10

Reconsider the accompanying sample data on expense ratio (%) for large-cap growth mutual funds first introduced in Exercise 1.53.

\(\begin{array}{l}0.52 1.06 1.26 2.17 1.55 0.99 1.10 1.07 1.81 2.05\\0.91 0.79 1.39 0.62 1.52 1.02 1.10 1.78 1.01 1.15\end{array}\)

A normal probability plot shows a reasonably linear pattern.

a. Is there compelling evidence for concluding that the population mean expense ratio exceeds \(1\% \)? Carry out a test of the relevant hypotheses using a significance level of \(.01\).

b. Referring back to (a), describe in context type I and II errors and say which error you might have made in reaching your conclusion. The source from which the data was obtained reported that \(\mu = 1.33\) for the population of all \(762\) such funds. So, did you actually commit an error in reaching your conclusion?

c. Supposing that \(\sigma = .5\), determine and interpret the power of the test in (a) for the actual value of m stated in (b).

It is known that roughly \(2/3\) of all human beings have a dominant right foot or eye. Is there also right-sided dominance in kissing behaviour? The article 鈥淗uman Behaviour: Adult Persistence of Head-Turning Asymmetry鈥� (Nature, 2003: 771) reported that in a random sample of \(124\) kissing couples, both people in \(80\) of the couples tended to lean more to the right than to the left.

a. If \(2/3\) of all kissing couples exhibit this right-leaning behaviour, what is the probability that the number in a sample of \(124\) who do so differ from the expected value by at least as much as what was actually observed?

b. Does the result of the experiment suggest that the \(2/3\) figure is implausible for kissing behaviour? State and test the appropriate hypotheses.

Answer the following questions for the tire problem in Example 8.7.

a.If \(\overline x = 30,960\) 30,960 and a level 伪=.01 test is used, what is the decision?

b.If a level .01 test is used, what is 尾(30,500)?

c.If a level .01 test is used and it is also required that 尾(30,500) = .05, what sample size n is necessary?

d.If \(\overline x = 30,960\), what is the smallest 伪 at which H0 can be rejected (based on n = 16)?
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