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Chapter 8: Q73SE (page 359)

In Problems 21鈥�24 verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution.

\(\frac{{dy}}{{dx}} + 4xy = 8{x^3};y = 2{x^2} - 1 + {c_1}{e^{ - 2{x^2}}}\)

Short Answer

Expert verified

The indicated function is a solution of the given differential equation for every real number \(x\).

Step by step solution

01

Determine the derivatives of the function.

Let the given function be \(y = 2{x^2} - 1 + {c_1}{e^{ - 2{x^2}}}\).

Then, the first derivative of the function is,

\(\frac{{dy}}{{dx}} = 4x - 4x{c_1}{e^{ - 2{x^2}}}\)

02

Determine the interval of the solution.

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

\(\begin{aligned}{c}\frac{{dy}}{{dx}} + 4xy &= 4x - 4x{c_1}{e^{ - 2{x^2}}} + 4x\left( {2{x^2} - 1 + {c_1}{e^{ - 2{x^2}}}} \right)\\ &= 4x - 4x{c_1}{e^{ - 2{x^2}}} + 8{x^3} - 4x + 4x{c_1}{e^{ - 2{x^2}}}\\ &= 8{x^3}\end{aligned}\)

That is same as the right-hand side of the differential equation for every real number \(x\). Thus, the indicated function is a solution of the given differential equation.

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Most popular questions from this chapter

A plan for an executive travelers鈥� club has been developed by an airline on the premise that \(5\% \) of its current customers would qualify for membership. A random sample of \(500\) customers yielded \(40\) who would qualify.

a. Using this data, test at level \(.01\) the null hypothesis that the company鈥檚 premise is correct against the alternative that it is not correct.

b. What is the probability that when the test of part (a) is used, the company鈥檚 premise will be judged correct when in fact \(10\% \) of all current customers qualify?

In Problems \(15 - 18\) verify that the indicated function \(y = \phi (x)\) is an explicit solution of the given first-order differential equation. Proceed as in Example \(6\), by considering \(\phi \) simply as a function and give its domain. Then by considering \(\phi \) as a solution of the differential equation, give at least one interval \(I\) of definition.

\(2y' = {y^3}cosx;y = {(1 - sinx)^{ - 1/2}}\)

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?

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

For each of the following assertions, state whether it is a legitimate statistical hypothesis and why:

\(\begin{array}{l}{\rm{a}}{\rm{. H: \sigma > 100}}\\{\rm{c}}{\rm{. H: s }} \le {\rm{.20}}\\{\rm{e}}{\rm{. H:}}\overline {{\rm{ X}}} {\rm{ - }}\overline {\rm{Y}} {\rm{ = 5}}\end{array}\) \(\begin{array}{l}{\rm{b}}{\rm{. H: }}\widetilde {\rm{x}}{\rm{ = 45}}\\{\rm{d}}{\rm{. H: }}{{\rm{\sigma }}_{\rm{1}}}{\rm{/}}{{\rm{\sigma }}_{\rm{2}}}{\rm{ < 1}}\end{array}\)

\({\rm{f}}{\rm{. H: \lambda }} \le {\rm{.01}}\), where \({\rm{\lambda }}\) is the parameter of an exponential distribution used to model component lifetime

In Problems \(15 - 18\) verify that the indicated functionis an explicit solution of the given first-order differential equation. Proceed as in Example \(6\), by considering \(\phi \) simply as a function and give its domain. Then by considering \(\phi \) as a solution of the differential equation, give at least one interval \(I\) of definition.

\(y' = 25 + {y^2};y = 5tan5x\)

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