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Chapter 12: Q26E (page 509)

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.

\(2x\frac{{dy}}{{dx}} - y = 2xcosx;y = \sqrt x \int_4^x {\frac{{cost}}{{\sqrt 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 = \sqrt x \int_4^x {\frac{{cost}}{{\sqrt t }}} dt\].

Multiply each side of the equation by\({x^{ - \frac{1}{2}}}\].

\(\begin{aligned}{c}y{x^{ - \frac{1}{2}}} = {x^{ - \frac{1}{2}}}{x^{\frac{1}{2}}}\int_4^x {\frac{{cost}}{{\sqrt t }}} \;dt\\y{x^{ - \frac{1}{2}}} = \int_4^x {\frac{{cost}}{{\sqrt t }}} \;dt\end{aligned}\]

02

Determine the solution of the indicated function.

Take differential on both sides of the equation.

Multiply\(2{x^{\frac{3}{2}}}\]on both sides of the equation.

\(\begin{aligned}{c}2x\frac{{dy}}{{\;dx}} - y = 2{x^{\frac{3}{2}}}{x^{ - \frac{1}{2}}}cosx\\2x\frac{{dy}}{{\;dx}} - y = 2xcosx\end{aligned}\]

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

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Most popular questions from this chapter

(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 27鈥�30 use (12) of Section 1.1 to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution.

\({x^2}y'' + \left( {{x^2} - x} \right)y' + (1 - x)y = 0;\;\;\;y = x\int_1^x {\frac{{{e^{ - t}}}}{t}} dt\)

Raindrops Keep Falling In meteorology the term virga refers to falling raindrops or ice particles that evaporate before they reach the ground. Assume that a typical raindrop is spherical. Starting at some time, which we can designate as t = 0, the raindrop of radius r0 falls from rest from a cloud and begins to evaporate.

(a) If it is assumed that a raindrop evaporates in such a manner that its shape remains spherical, then it also makes sense to assume that the rate at which the raindrop evaporates鈥攖hat is, the rate at which it loses mass鈥攊s proportional to its surface area. Show that this latter assumption implies that the rate at which the radius r of the raindrop decreases is a constant. Find r(t). (Hint: See Problem 55 in Exercises 1.1.)

(b) If the positive direction is downward, construct a mathematical model for the velocity v of the falling raindrop at time t > 0. Ignore air resistance. (Hint: Use the form of Newton鈥檚 second law of motion given in (17).)

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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