/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q53E Verify that the piecewise-define... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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 \(( - \infty ,\infty )\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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.

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.

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

Reconsider the paint-drying situation of Example 8.5, in which drying time for a test specimen is normally distributed with 蟽 = 9. The hypotheses H0: 碌 =75 versus Ha: 碌 <75 are to be tested using a random sample of n= 25 observations.

a.How many standard deviations (of X) below the null value is \(\overline x = 72.3\)?

b.If \(\overline x = 72.3\), what is the conclusion using 伪 =.002?

c.For the test procedure with 伪 =.002, what is 尾(70)?

d.If the test procedure with 伪 =.002 is used, what n is necessary to ensure that 尾(70) = .01?

e.If a level .01 test is used with n5 100, what is the probability of a type I error when m5 76?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)?

Automatic identification of the boundaries of significant structures within a medical image is an area of ongoing research. The paper 鈥淎utomatic Segmentation of Medical Images Using Image Registration: Diagnostic and

Simulation Applications鈥� (J. of Medical Engr. and Tech., 2005: 53鈥�63)discussed a new technique for such identification. A measure of the accuracy of the automatic region is the average linear displacement (ALD). The paper gave the following ALD observations for a sample of 49 kidneys (units of pixel dimensions).

1.38 0.44 1.09 0.75 0.66 1.28 0.51

0.39 0.70 0.46 0.54 0.83 0.58 0.64

1.30 0.57 0.43 0.62 1.00 1.05 0.82

1.10 0.65 0.99 0.56 0.56 0.64 0.45

0.82 1.06 0.41 0.58 0.66 0.54 0.83

0.59 0.51 1.04 0.85 0.45 0.52 0.58

1.11 0.34 1.25 0.38 1.44 1.28 0.51

a.Summarize/describe the data.

b.Is it plausible that ALD is at least approximately normally distributed? Must normality be assumed prior to calculating a CI for true average ALD or testing hypotheses about true average ALD? Explain.

c.The authors commented that in most cases the ALD is better than or of the order of 1.0. Does the data in fact provide strong evidence for concluding that true average ALD under these circumstances is less than 1.0? Carry out an appropriate test of hypotheses.

d.Calculate an upper confidence bound for true average ALD using a confidence level of 95%, and interpret this bound.

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

See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.