Chapter 8: Ordinary Differential Equations

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

$\mathit{y}\mathbf{"}\mathbf{-}\mathbf{4}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathit{y}\mathbf{=}\mathbf{4}$, $\mathbf{\sum }_{\mathbf{0}}\mathbf{=}\mathbf{0}\mathbf{,}{\mathit{\psi }}_{\mathbf{0}}\mathbf{=}\mathbf{-}\mathbf{2}$

Use the convolution integral to find the inverse transforms of:$\frac{\mathbf{p}}{\left(p+a\right)\left(p^2+b^2\right)}$

In problems 13 to 15, find a solution (or solutions) of the differential equation not obtainable by specializing the constant in your solution of the original problem. Hint: See Example 3

Problem 11

By separation of variables, find a solution of the equation $\mathit{y}\mathbf{\text{'}}\mathbf{=}\sqrt{\mathbf{y}}$ containing one arbitrary constant. Find a particular solution satisfying y=0 when x=0. Show that y=0 is a solution of the differential equation which cannot be obtained by specializing the arbitrary constant in your solution above. Computer plot a slope field and some of the solution curves. Show that there are an infinite number of solution curves passing through any point on the xaxis, but just one through any point for which y>0 . Hint:See Example 3. Problems 17 and 18 are physical problems leading to this differential equation.

Let the rate of growth $\frac{\mathbf{d}\mathbf{N}}{\mathbf{d}\mathbf{t}}$ of a colony of bacteria be proportional to the square root of the number present at any time. If there are no bacteria present at t=0 , how many are there at a later time? Observe here that the routine separation of variables solution gives an unreasonable answer, and the correct answer, N=0 , is not obtainable from the routine solution. (You have to think, not just follow rules!)

Consider the following special cases of the simple series circuit [Figure1.1 and equation ( 1.2)].

(a) RC circuit (that is, L=0 ) with V=0 ; find q as a function of t if ${\mathit{q}}_{\mathbf{0}}$ is the charge on the capacitor at t=0 .

(b) RL circuit (that is, no capacitor; this means$\frac{\mathbf{1}}{\mathbf{C}}\mathbf{=}\mathbf{0}$) with V=0 ; find $\mathit{I}\left(t\right)$ given $\mathit{I}\mathbf{=}{\mathit{I}}_{\mathbf{0}}\mathbf{}\mathit{a}\mathit{t}\mathbf{}\mathit{t}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$.

(c) Again note that these are the same differential equations as in Problem 19 and Example 1. The terminology is again different; we define the time constant $\tau$ for a circuit as the time required for the charge (or current) to fall to role="math" localid="1664336989028" $\frac{\mathbf{1}}{\mathbf{e}}$times its initial value. Find the time constant for the circuits (a) and (b) . If the same equation, say role="math" localid="1664336998228" $\mathit{y}\mathbf{=}{\mathit{y}}_{\mathbf{0}}{\mathit{e}}^{\mathbf{-}\mathbf{a}\mathbf{t}}$ , represented either radioactive decay or light absorption or an RC or RL circuit, what would be the relations among the half-life, the half-value thickness, and the time constant?

Show that the thickness of the ice on a lake increases with the square root of the time in cold weather, making the following simplifying assumptions. Let the water temperature be a constant$\mathbf{10}\mathbf{°}\mathit{C}$, the air temperature a constant$\mathbf{-}\mathbf{10}\mathbf{°}\mathit{C}$, and assume that at any given time the ice forms a slab of uniform thickness x. The rate of formation of ice is proportional to the rate at which heat is transferred from the water to the air. Let t=0when x=0.

An object of mass falls from rest under gravity subject to an air resistance proportional to its speed. Taking the yaxis as positive down, show that the differential equation of motion is$\mathit{m}\left(dv/dt\right)\mathbf{=}\mathit{m}\mathit{g}\mathbf{-}\mathit{k}\mathit{V}$ , where kis a positive constant. Find vas a function of t , and find the limiting value of vas tends t to infinity; this limit is called the terminal speed. Can you find the terminal speed directly from the differential equation without solving it?

Hint:What is $\mathit{d}\mathit{v}\mathbf{/}\mathit{d}\mathit{t}$after vhas reached an essentially constant value?

Consider the following specific examples of this problem.

(a) A person drops from an airplane with a parachute. Find a reasonable valueOf k .

(b) In the Millikan oil-drop experiment to measure the charge of an electron, tiny electrically charged drops of oil fall through the air under gravity or rise under the combination of gravity and an electric field. Measurements can be made only after they have reached terminal speed. Find a formula for the time required for a drop starting at rest to reach $\mathbf{99}\mathbf{\%}$of its terminal speed.

According to Newton’s law of cooling, the rate at which the temperature of an object changes is proportional to the difference between its temperature and that of its surroundings. A cup of coffee at 200° in a room of temperature 70^° is stirred continuously and reaches 100° after 10 min. At what time was it at 120°?

A glass of milk at $\mathbf{38}\mathbf{°}$is removed from the refrigerator and left in a room at a temperature $\mathbf{70}\mathbf{°}$^°. If the temperature of the milk is$\mathbf{54}\mathbf{°}$ after 10min , what will its temperature be in half an hour? (See Problem 27.)