Math 23
Exam 1 Practice Problems

1.

Solve the following initial value problem:

$$y' + yy' = e^t \qquad \qquad y(0)=-4$$

2.

Consider the general first order linear differential equation

$$y' = -ry, \quad r > 0$$ (1)

(a)

State the general solution to the equation

(b)

What can you say about the behavior of solutions as $t \to \infty$?

3.

Consider the DE

y'' - y' - 2y = x. (2)

(a)

Find a solution to Equation () which is a linear function.

(b)

Find the general solution to the homogeneous equation.

4.

Find two linearly independent solutions to the differential equation

y'' - 25y = 0. (3)

5.

Solve the initial value problem

$$y'' - 5y' + 6y = 0, \quad y(0) = 1, \quad y'(0) = -1$$ (4)

6.

Find the limit of the sequence $a_n = \frac{1}{n^2 + 1}$, and prove that it exists using the definition of convergence.

7.

(a)

Compute the Taylor series for f(x) = e^x2 about x = 0.

(b)

Compute the Taylor series for $f(x) =\ln{(5x-4)}$ about x = 1, and find the radius of convergence.

8.

(a)

Find the Taylor series for $\quad f(t) = t^2 + 3t \quad$ about t=0.

(b)

Prove that if $\quad f(t)=a_nt^n + a_{n-1}t^{n-1} + \cdots + a_2t^2 + a_1t+a_0 \quad$ is any polynomial, then the Taylor series for f about t=0 is f itself.

9.

State whether each of the following series converges or diverges. You do not need to give an explanation. You need not actually find the sum.

(a)

$${\sum_{n=0}^{\infty} \frac{1}{2^n}}$$

(b)

$${\sum_{n=0}^{\infty} \frac{2^{n+3}}{3^{n+2}}}$$

(c)

$${\sum_{n=1}^{\infty} \frac{1}{10\sqrt{n}}}$$

(d)

$${\sum_{n=1}^{\infty} \frac{1}{(1.1)^n \sqrt{n}}}$$

(e)

$${\sum_{n=1}^{\infty} \frac{n^5 + 4n^4 - 1}{n^{10} - 17n^2}}$$

(f)

$${\sum_{n=1}^{\infty} \frac{1+\frac{1}{n}}{1+\frac{1}{n^2}}}$$

(g)

$${\sum_{n=1}^{\infty} \frac{(-1)^n}{n}}$$

(h)

$${\sum_{n=3}^{\infty} \frac{1}{(n-1)(n-2)}}$$

(i)

$${\sum_{n=1}^{\infty} \frac{1}{(2n+2)(n+2)}}$$

10.

At time t=0, a 1000 gallon tank is full of a 5% potassium chloride (KCl) solution. The tank is to be rinsed by having clean water flow into the tank at the rate of 15 gallons/min, and the well-mixed solution flowing out at the same rate.

(a)

Write down the differential equation governing the concentration of KCl in the tank at time t.

(b)

How long will it take the concentration of KCl in the tank to reach .1%?

(c)

What is the minimum input/output flow rate of fresh water that will allow the tank mixture to drop from 5% to .1% in one hour?

11.

An object of mass m falling near the surface of the earth is retarded by air resistance proprotional to its speed. Let v(t) denote the object's velocity at time t, and let g denote the (constant) acceleration due to gravity.

(a)

Verify that this DE has exactly one equilibrium solution. Find it, and state whether it is asymptotically stable or unstable. Explain (pictures may help).

(b)

What is the physical meaning of the equilibrium solution you found above? I.e. what does it say about the speed of a falling object near the earth's surface?

12.

Recall that Newton's law of cooling states that the rate of change of the temperature of an object is proportional to the difference between the temperature of the object and the temperature of the surrounding environment. Assume that, for a certain object, the constant of proportionality is k=-0.2 (Assume that units on k are 1/min.). Now assume that that object is at 150 degrees when you place it into a room whose temperature is oscillating according to the function $T(t) = 70+5\cos t$ (where t represents minutes after the object was placed in the room).

Find the temperature of the object at any time $t \geq 0$.

HELPFUL NOTE: for all a, $\int e^{at} \cos t dt = \frac{e^{at}}{a^2+1}(a \cos t + \sin t) + C$