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Chapter 12: Q21E (page 508)

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

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.

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

(a) Verify that the one-parameter family \({y^2} - 2y = {x^2} - x + c\) is an implicit solution of the differential equation \((2y - 2)y' = 2x - 1\).

(b) Find a member of the one-parameter family in part (a) that satisfies the initial condition \(y(0) = 1\).

(c) Use your result in part (b) to and an explicit function \(y = \phi (x)\) that satisfies \(y(0) = 1\). Give the domain of the function \(\phi \). Is \(y = \phi (x)\) a solution of the initial-value problem? If so, give its interval \(I\) of definition; if not, explain.

A tank in the form of a right-circular cylinder of radius \(2\) feet and height \(10\) feet is standing on end. If the tank is initially full of water and water leaks from a circular hole of radius \(12\) inch at its bottom, determine a differential equation for the height h of the water at time \(t > 0\). Ignore friction and contraction of water at the hole.

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.

\(\frac{{dP}}{{dt}} = P(1 - P);\;P = \frac{{{c_1}{e^t}}}{{1 + {c_1}{e^t}}}\)

Population Model The differential equation \(\frac{{dP}}{{dt}} = (kcost)\), where \(k\) is a positive constant, is a model of human population \(P(t)\) of a certain community. Discuss an interpretation for the solution of this equation. In other words, what kind of population do you think the differential equation describes?
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