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Chapter 8: Q77SE (page 359)

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.

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

Short Answer

Expert verified

The indicated function is a solution of the differential function.

Step by step solution

01

Simplify the given differential equation.

Let the given differential equation be \(y = {e^{3x}}\int_1^x {\frac{{{e^{ - 3t}}}}{t}} dt\).

Multiply each side of the equation by \({e^{ - 3x}}\).

\(\begin{array}{l}y{e^{ - 3x}} = {e^{3x}}{e^{ - 3x}}\int_1^x {\frac{{{e^{ - 3t}}}}{t}} \;dt\\y{e^{ - 3x}} = \int_1^x {\frac{{{e^{ - 3t}}}}{t}} \;dt\end{array}\)

02

Determine the solution of the indicated function.

Take differential on both sides of the equation.

Multiply \(x\) on both sides of the equation.

\(x{e^{ - 3x}}\frac{{dy}}{{\;dx}} - 3yx{e^{ - 3x}} = {e^{ - 3x}}\)

Divide\({e^{ - 3x}}\)on both sides of the equation.

\(x\frac{{dy}}{{\;dx}} - 3yx = 1\)

Hence, the indicated function is a solution of the differential function.

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Most popular questions from this chapter

A manufacturer of plumbing fixtures has developed a new type of washer less faucet. Let \(p = P\) (a randomly selected faucet of this type will develop a leak within \(2\) years under normal use). The manufacturer has decided to proceed with production unless it can be determined that \(p\) is too large; the borderline acceptable value of \(p\) is specified as \(.10\). The manufacturer decides to subject \(n\) of these faucets to accelerated testing (approximating \(2\) years of normal use). With \(X = \) the number among the \(n\) faucets that leak before the test concludes, production will commence unless the observed X is too large. It is decided that if \(p = .10\), the probability of not proceeding should be at most \(.10\), whereas if \(p = .30\) the probability of proceeding should be at most \(.10\). Can \(n = 10\) be used? \(n = 20\)? \(n = 25\)? What are the actual error probabilities for the chosen n?

A random sample of \(150\) recent donations at a certain blood bank reveals that \(82\) were type A blood. Does this suggest that the actual percentage of type A donations differs from \(40\% \), the percentage of the population having type A blood? Carry out a test of the appropriate hypotheses using a significance level of \(.01\). Would your conclusion have been different if a significance level of \(.05\) had been used?

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

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

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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