Consider the equation $y”−y'−2y=0$. (a) Show that $y_1(t)=e^{−t}$ and $y_2(t)=e^{2t}$ form a fundamental set of solutions. (b) Let $y_3(t)=−2e^{2t}$, $y_4(t)=y_1(t)+2_{y2}(t)$, and $y_5(t)=2_{y1}(t)−2_{y3}(t)$. Are $y_3(t)$, $y_4(t)$, and $y_5(t)$ also solutions of the given differential equation? (c) Determine whether each of the following pairs forms a fundamental set of solutions: $[y_1(t),y_3(t)];[y_2(t),y_3(t)];[y_1(t),y_4(t)];[y_4(t),y_5(t)]$.

Convert the third-order differential equation

$$y'''-2y''-y'+2y=0$$

into a system of differential equations, and then find the particular solution such that $y(0)=2$, $y'(0)=-3$, and $y''(0)=5$.

Let

$$Y _ { 1 } , Y _ { 2 } , \ldots , Y _ { 5 }$$

be a random sample of size 5 from a normal population with mean 0 and variance 1 and let

$$\overline { Y } = ( 1 / 5 ) \sum _ { i = 1 } ^ { 5 } Y _ { i }$$

. Let

$$Y _ { 6 }$$

be another another independent observation from the same population. What is the distribution of

$$W = \sum _ { i = 1 } ^ { 5 } Y _ { i } ^ { 2 }$$

? Why?

Suppose water is leaking from a cubical tank through a circular hole of area $A_h$ at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of the water leaving the tank per second to $cA_h \sqrt{2gh}$, where $c (0 < c < 1)$is an empirical constant. Determine a differential equation for the height $h$ of water at time $t$. The radius of the hole is 2 in and $g$ = 32 ft/s$^{\text{2}}$.

For the differential equation

$$\begin{gather*} y' = \dfrac{2x}{1 + 2y} \end{gather*}$$

with the initial value y(2) = 0, (a) Find the solution in explicit form. (b) Plot the graph of the solution. (c) Determine (at least approximately) the interval in which the solution is defined.

Consider the differential equation ay''+by'+cy=g(t),(i)where a,b, and c are positive. If Y1(t) and Y2(t)are solutions of Eq. (i), show that Y1(t)−Y2(t)→0 as t→∞. Is this result true if b=0?

Consider the initial value problem y'+14y=3+2 cos 2t, y(0)=0.(a) Find the solution of this initial value problem and describe its behavior for large t.(b) Determine the value of t for which the solution first intersects the line y=12.

Two chemicals $A$ and $B$ are combined to form a chemical $C$. The rate, or velocity, of the reaction is proportional to the product of the instantaneous amounts of $A$ and $B$ not converted to chemical $C$. Initially there are 100 grams of $A$ and 50 grams of $B$, and for each gram of $B$, 2 grams of $A$ is used. It is observed that 10 grams of $C$ is formed in 5 minutes. How much is formed in 20 minutes? What is the limiting amount of $C$ after a long time? How much of chemicals $A$ and $B$ remains after a long time? At what time is chemical $C$ half-formed?

(a) A simple model for the shape of a tsunami is given by $\frac{d W}{d x}=W \sqrt{4-2 W}$, where W(x) > 0 is the height of the wave expressed as a function of its position relative to a point offshore . By inspection , find all constant solution s of the DE. (b) Solve the differential equation in part (a). A CAS may be useful for integration. (c) Use a graphing utility to obtain the graphs of all solutions that satisfy the initial condition W(0) = 2.

Verify that y1(t)=1 and y2(t)=t1/2 are solutions of the differential equation yy”+(y')2=0 for t>0. Then show that y=c1+c2t1/2 is not, in general, a solution of this equation.

Suppose a student carrying a flu virus returns to an isolated college campus of 1000 students. Determine a differential equation for the number of people x(t) who have contracted the flu if the rate at which the disease spreads is proportional to the number of interact ions between the numb er of students who have the flu and the numb er of students who have not yet been exposed to it.

(a) A simple model for the shape of a tsunami is given by

$$\frac { d W } { d x } = W \sqrt { 4 - 2 W },$$

where $W(x) > 0$ is the height of the wave expressed as a function of its position relative to a point offshore. By inspection, find all constant solutions of the DE. (b) Solve the differential equation in part (a). A CAS may be useful for integration. (c) Use a graphing utility to obtain the graphs of all solutions that satisfy the initial condition $W(0) = 2$.

Suppose a student carrying a flu virus returns to an isolated college campus of 1000 students. Determine a differential equation governing the number of students $x(t)$ who have contracted the flu if the rate at which the disease spreads is proportional to the number of interactions between the number of students with the flu and the number of students who have not yet been exposed to it.

Determine the longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution. Do not attempt to find the solution.

y''+(cost)y'+3(ln|t|)y=0, y(2)=3, y'(2)=1