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

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

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 .

Solve for l(t), assuming that R = 500 ohms, L = 4 henries, and V = 20 cos(80) volts.

Solve the Initial Value Problem.

$$y ^ { \prime } = y ^ { 2 } \sin x , \quad y ( \pi ) = 2$$

At time t = 0, a tank of height 5 m in the shape of an inverted pyramid whose cross section at the top is a square of side 2 m is filled with water. Water flows through a hole at the bottom of area

$$0.002 \mathrm { m } ^ { 2 }.$$

Use Torricelli's Law to determine the time required for the tank to empty.

Let A and B be constants. Prove that if A > 0, then all solutions of

$$\frac { d y } { d t } + A y = B$$

approach the same limit as

$$t \rightarrow \infty.$$

Solve the Initial Value Problem. y' = tan y,

$$y ( \ln 2 ) = \frac { \pi } { 2 }$$

Assume that V(t) = V is constant and I(0) = 0. (a) Solve for I(t). (b) Show that

$$\lim _ { t \rightarrow \infty } I ( t ) = V / R$$

and that I(t) reaches approximately 63% of its limiting value after L/R seconds. (c) How long does it take for I(t) to reach 90% of its limiting value if R = 500 ohms, L = 4 henries, and V = 20 volts?

Uranium-238 is a radioactive material with a half-life of 4.468 billion years. With t in billions of years, let M(t) be the mass in grams of a sample of uranium-238 that initially consisted of 100 g. Set up and solve an Initial Value Problem for determining M(t).

In this exercise, consider a series circuit consisting of a resistor of R ohms, an inductor of L henries, and a variable voltage source of V(t) volts (time t in seconds). The current through the circuit I(t) (in amperes) satisfies the differential equation

$$\frac{d I}{d t}+\frac{R}{L} I=\frac{1}{L} V(t)$$

Assume that R=110 ohms, L=10 henries, and V(t)= e$^{-t}$ volts.

(a) Solve the equation above with initial condition $I(0)=0$.

(b) Calculate $t_m$ and $I\left(t_m\right)$, where $t_m$ is the time at which $I(t)$ has a maximum value.

(c) Use a computer algebra system to sketch the graph of the solution for $0 \leq t \leq 3$.

Solve the Initial Value Problem.

$$\sqrt { 1 - x ^ { 2 } } y ^ { \prime } = y ^ { 2 } + 1,$$

y(0) = 0

For what function P is the integrating factor $\alpha(x)$ equal to e$^x$?