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Chapter 12: Q3E (page 494)
In Problems \(1 - 8\) state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with \((6)\).
\({t^5}{y^{(4)}} - {t^3}y'' + 6y = 0\)
The equation is linear and fourth order.
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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.
\((y - 1)y' = 1\)
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).)
In Problems \(31 - 34\) find values of m so that the function \(y = m{e^{mx}}\) is a solution of the given differential equation.
\(y' + 2y = 0\)
A differential equation may possess more than one family of solutions.
(a) Plot different members of the families \(y = {\phi _1}(x) = {x^2} + {c_1}\) and \(y = {\phi _2}(x) = - {x^2} + {c_2}\).
(b) Verify that \(y = {\phi _1}(x)\) and \(y = {\phi _2}(x)\) are two solutions of the nonlinear first-order differential equation \({(y')^2} = 4{x^2}\).
(c) Construct a piecewise-defined function that is a solution of the nonlinear DE in part (b) but is not a member of either family of solutions in part (a).
Radioactive Decay Suppose that \(dA/dt = - 0.0004332A(t)\) represents a mathematical model for the decay of radium-, whereis the amount of radium (measured in grams) remaining at time(measured in years). How much of the radium sample remains at the time when the sample is decaying at a rate ofgrams per year?\(226\)
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