In this exercise, let $\alpha(x)$ be an integrating factor for $y^{\prime}+P(x) y=Q(x)$. The differential equation $y^{\prime}+P(x) y=0$ is called the associated homogeneous equation.

(a) Show that $y=1 / \alpha(x)$ is a solution of the associated homogeneous equation.

(b) Show that if $y=f(x)$ is a particular solution of $y^{\prime}+P(x) y=Q(x)$, then $f(x)+C / \alpha(x)$ is also a solution for any constant $C$.

Suppose that y' = ky(1 - y/8) has a solution satisfying y(0) = 12 and y(10) = 24. Find k.

Assume that the outside temperature varies as

T(t)=15+5 sin ($\pi$ t / 12)

where $t=0$ is 12 noon. A house is heated to $25^{\circ}$C at t=0 and after that, its temperature y(t) varies according to Newton's Law of Cooling :

dy/dt=-0.1(y(t)-T(t))

Use Earlier Exercise to solve for y(t).

In Exercise, use the differential equation for a leaking container, Eq. $\frac{d y}{d t}=-\frac{B \sqrt{2 g y}}{A(y)}$

Water leaks through a hole of area $B=0.002 \mathrm{~m}^2$ at the bottom of a cylindrical tank that is filled with water and has height 3 m and a base of area $10 \mathrm{~m}^2$. How long does it take (a) for half of the water to leak out and (b) for the tank to empty ?

Let a, b, r be constants. Show that

$$y = C e ^ { - k t } + a + b k \left( \frac { k \sin rt - r \cos r t } { k ^ { 2 } + r ^ { 2 } } \right)$$

is a general solution of

$$\frac { d y } { d t } = - k ( y - a - b \sin r t )$$

Let y(t) be the solution of

$$\frac { d y } { d t } = 0.3 y ( 2 - y )$$

with y(0) = 1. Determine

$$\lim _ { t \rightarrow \infty } y ( t )$$

without solving for y explicitly.

In this exercise, let y(t) be the solution of dy/dt=0.3y(2-y) with y(0)=1. Determine $\lim \limits_{t \to \infty}$ y(t) without solving for y explicitly.

In this exercise, let a, b, r be constants. Show that

$$y=C e^{-k t}+a+b k\left(\frac{k \sin r t-r \cos r t}{k^2+r^2}\right)$$

is a general solution of

$$\frac{d y}{d t}=-k(y-a-b \sin r t)$$

In the earlier example, we calculated the thickness of a glacier assuming the friction at the base of the glacier is constant. In this exercise, we consider cases where the friction varies along the length of the glacier.

(a) Solve the glacier thickness differential equation, and use the equation

$$T \frac{d T}{d x}=\frac{\tau}{\rho g}$$

for T(x) with $\tau(x)=0.3 x(1000-x)$ N/m$^2$ and initial condition T(0)=0.

(b) Sketch the graph of T for $0 \leq x \leq 1000$ m.

In this exercise, find the solutions of the logistic equation dy/dt=y(4-y) satisfying the initial conditions:

(a) y(0)=1

(b) y(0)=4

(c) y(0)=6

Continuing with the previous exercise, let tank 2 be another tank filled with V$_2$ gallons of water. Assume that the dye solution from tank 1 empties into tank 2 as in the previous figure, mixes instantaneously, and leaves tank 2 at the same rate R. Let c$_2$(t) be the dye concentration in tank 2 at time t.

(a) Explain why c$_2$ satisfies the differential equation

$$\frac{d c_2}{d t}=\frac{R}{V_2}\left(c_1-c_2\right)$$

(b) Use the solution to the previous exercise to solve for $c_2(t)$ if $V_1=300$, $V_2=200, R=50$, and $c_0=10$.

(c) Find the maximum concentration in tank 2.

(d) Plot the solution.

Tank 1 is filled with $V_1$ liters of water containing blue dye at an initial concentration of $c_0 \mathrm{~g} / \mathrm{L}$. Water flows into the tank at a rate of RL /min, is mixed instantaneously with the dye solution, and flows out through the bottom at the same rate R. Let $c_1(t)$ be the dye concentration in the tank at time t.

(a) Explain why $c_1$ satisfies the differential equation

$$\frac{d c_1}{d t}=-\frac{R}{V_1} c_1$$

(b) Solve for $c_1(t)$ with $V_1=300 \mathrm{~L}$, R=50, and $c_0=10 \mathrm{~g} / \mathrm{L}$.

In the earlier example, we calculated the thickness of a glacier assuming the friction at the base of the glacier is constant. In this exercise, we consider cases where the friction varies along the length of the glacier.

(a) Solve the glacier thickness differential equation, and use the equation below

$$T \frac{d T}{d x}=\frac{\tau}{\rho g}$$

for T(x) with $\tau(x)=75 x$ N/m$^2$ and initial condition T(0)=0.

(b) Sketch the graph of T for $0 \leq x \leq 1000$ m.

The trough (dimensions in centimeters) is filled with water. At time t=0 (in seconds), water begins leaking through a hole at the bottom of area 4$cm^2$ . Let y(t) be the water height at time t . Find a differential equation for y(t) and solve it to determine when the water level decreases to 60 cm .