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Chapter 12: Q37E (page 518)
In Problems \(37 - 40\) use the concept that \(y = c, - \infty < x < \infty \), is a constant function if and only if \(y' = 0\) to determine whether the given differential equation possesses constant solutions.
\(3xy' + 5y = 0\)
There exists a constant solution (i.e.) \(y = 2\).
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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?
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 \(\phi \).
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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