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Chapter 8: Q68SE (page 358)

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.

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

Short Answer

Expert verified

The indicated function is an explicit solution of the given differential equation and the interval is \(I:\left( { - \frac{\pi }{{10}},\frac{\pi }{{10}}} \right)\), and the domain is \(D:x \ne \frac{\pi }{{10}} + \frac{{n\pi }}{5}\).

Step by step solution

01

Define an explicit function.

Anexplicit solutionis one in which the dependent variable is expresseddirectlyin terms of the independent variable and constants.

Let the given function be .

Then, the first derivative of the function is,

\(y' = 25se{c^2}5x\)

02

Determine the explicit solution.

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

\(\begin{aligned}{c}25se{c^2}5x &= 25 + 25ta{n^2}5x\\25se{c^2}5x &= 25\left( {1 + ta{n^2}5x} \right)\\25se{c^2}5x &= 25se{c^2}5x\end{aligned}\)

That is same as the right-hand side of the differential equation. The indicated function is an explicit solution of the given differential equation.

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Most popular questions from this chapter

In Problems \(31 - 34\) find values of m so that the function \(y = m{e^{mx}}\) is a solution of the given differential equation.

\(5y' = 2y\)

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.

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

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.

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

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.

In Problems \(19\) and \(20\) verify that the indicated expression is an implicit solution of the given first-order differential equation. Find atleast one explicit solution \(y = \phi (x)\) in each case. Use a graphing utility to obtain the graph of an explicit solution. Give an interval \(I\) of definition of each solution \(\phi \).
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