Exam 1 Review Worksheet

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1. Verify that x = te−t is a solution of x′′ + x = −2x′.

2. Verify that x = c1 cos 3t + c2 sin 3t is a solution of x′′ + 9x = 0.

3. Verify that x = c1t + c2t3 is a solution of t2x′′ = 3tx′ − 3x

4. Match the differential equation with the direction field. Not all equations will be used.

(a) y′ = 2y − 1

(b) y′ = 2 + y

(c) y′ = y − 2

(d) y′ = y(y + 3)

(e) y′ = y(y − 3)

(f) y′ = 1 + 2y

(g) y′ = −2 − y

(h) y′ = y(3 − y)

(i) y′ = 1 − 2y

(j) y′ = 2 − y

(a) (b)

(c) (d)

(e) (f)

5. Solve the following differential equations.

(a)dy

dx=

ay + b

cy + d, a, b, c, d constants

(b) y′ + 1t y = 3 cos 2t, t > 0

(c) (yexy cos 2x− 2exy sin 2x + 2x) + (xexy cos 2x− 3)y′ = 0

(d)dy

dx=

x− e−x

y + ey

(e)x

(x2 + y2)3/2+

y

(x2 + y2)3/2dy

dx= 0

(f) ty′ + (t + 1)y = t, y(ln 2) = 1, t > 0

(g) y′ = xy3(1 + x2)−1/2, y(0) = 1

(h) (3x2 − 2xy + 2) + (6y2 − x2 + 3)y′ = 0

(i) y′ + y = 5 sin 2t

(j) y′ =3x2 − ex

2y − 5

(k) y′ + 2y = te−2t, y(1) = 0

(l) y′ +3y

5 − t= 1

(m) (9x2 + y − 1) − (4y − x)y′ = 0, y(1) = 0

(n) y′ +y

t ln t=

1

t

(o) (y/x + 6x) + (lnx− 2)y′ = 0, x > 0

6. Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains200 L of a dye solution with a concentration of 1 g/L. To prepare for the next experiment, the tankis to be rinsed with fresh water flowing in at a rate of 2L/min, the well-stirred solution flowing out atthe same rate. Find the time that will elapse before the concentration of the dye in the tank reaches1% of its original value.

7. A young person with no initial capital invests k dollars per year at an annual rate of return r. Assumethat the investments are made continuously and that the return is compounded continuously.

(a) Determine the sum S(t) accumulated at any time t.

(b) If r = 7.5%, determine k so that $1 million will be available for retirement in 40 years.

(c) If k = $2000/year, determine the return rate r that must be obtained to have $1 million availablein 40 years.

8. A rabbit population doubles every 6 months. If the colony starts with 500 rabbits, how long will ittake to reach 1500, assuming exponential growth?

9. A $5000 investment is made for 30 years at 8% annual interest. How much will the investment beworth at the end of the 30 years?

10. A 100 gallon tank holds 50 gallons of water with 75 grams of salt dissolved into it. If a mixture with 6grams per gallon of salt is pumped in at 2 gallons per hour and the mixture is drained at 1 gallon perhour, determine how much salt will be in the tank when it is full.

11. A murder victim is discovered at midnight and the temperature of the body is recorded at 31◦C. Onehour later, the temperature of the body is 29◦C. Assume that the surrounding air temperature remainsconstant at 21◦C. Calculate the victim’s time of death. Note: The normal body temperature of a livinghuman being is approximately 37◦C.

12. Suppose a cold beer at 40◦F is placed into a warm room at 70◦F. Suppose 10 minutes later, thetemperature of the beer is 48◦F. Use Newton’s law of cooling to find the temperature 25 minutes afterthe beer was placed into the room.

13. The isotope Technitium 99m is used in medical imaging. It has a half-life of about 6 hours, a usefulfeature for radioisotopes that are injected into humans. The Technitium, having such a short half-life,is created artificially on scene by harvesting it from a more stable Molybdenum isotope, 99Mb. If 10 gof 99mTc are ”harvested” from the Molybdenum, how much of the sample remains after 9 hours?

14. In the following problems:

(a) Find approximate values of the solution of the given initial value problem at t = 0.1, 0.2, 0.3, and0.4 using Euler’s method with h = 0.1.

(b) Repeat part (a) with h = 0.05. Compare the results with those found in (a).

(c) Repeat part (a) with h = 0.025. Compare the results with those found in (a) and (b).

(d) Find the solution to the differential equation, and compare the exact values of the solution fort = 0.1, 0.2, 0.3, and 0.4. Compare these values with the results of (a), (b), (c).

i. y′ = 3 + t− y, y(0) = 1.

ii. y′ = 2y − 1, y(0) = 1.

iii. y′ = 0.5 − t + 2y, y(0) = 1.

iv. y′ = 3 cos t− 2y, y(0) = 0