Mth 256 Test 1 Name: ID: Rec: 1300Bent Petersen 256w2004-test-1.tex February 12, 2004 Time: 50 minutes.Instructions: =⇒If you do not read the in-structions, then how willyou know what to do?Read them now.Be sure to write yourname in the space above.• You may use one 8.5 × 11 inch note sheet prepared in advance. You may write on bothsides of your note sheet.• Note sheets may not be shared. If you do not bring a note sheet you will have to dowithout any help notes.• You may not use any books, notebooks, additional note sheets nor note cards.• You are expected to have a simple scientific calculator available for use on this test.Calculators and other equipment may not be shared.• For multiple-choice problems place the letter corresponding to your answer in the boxprovided.• Note that log(x) means the natural logarithm of x.Problem 1. (10 points if correct, -1 points if wrong). Solve the initial value problemdydt+1ty = t2, y(1) = 1.Then choose the number in the list below closest to y(2).A.) 1.000 B.) 3.000C.) 2.375 D.) 3.125 E.) 3.250←Letter corresponding to your answer to problem 1.Problem 2. (10 points if correct, -1 points if wrong). Solve the initial value problemdydt+ y2= −y, y(0) = 3.A.) 2e−t+ 1 B.)23et−2C.)34et−3D.)12et−1E.) None of the above.←Letter corresponding to your answer to problem 2.Problem 3. (10 points if correct, -1 points if wrong). Solve the exact ordinary differential equation−4e−2xcos(3y) + 3 +−6e−2xsin(3y) − 4dydx= 0.A.) Not exact! B.) 2e−2xcos(3y) + 3x − 4y = cC.) 4e−2xcos(3y) + 3y − 4x = c D.) 4e−2xcos(3y) + 12xy = c E.) None of the above.←Letter corresponding to your answer to problem 3.Problem 4. (10 points if correct, -1 points if wrong). The ordinary differential equation4y + 15xy2− 49x3y +3x + 12yx2− 21x4dydx= 0has an integrating factor of the form µ = xmyn. Find m and n.A.) m = 3, n = 4 B.) m = 2, n = −3C.) m = 3, n = 2 D.) m = 2, n = 3 E.) None of the above.←Letter corresponding to your answer to problem 4.Problem 5. (10 points if correct, -1 points if wrong). A 100 gallon tank initially contains 50 gallons of brine ofconcentration 1.5 oz salt per gallon. Brine of concentration 3 oz salt per gallon flows into the tank at 4 gallons perminute. The well-mixed solution is pumped out at 2 gallons per minute. Find the concentration of salt in the brine inthe tank at the very moment of overflow. Select the closest number from the list below.A.) 2.625 B.) 3.175C.) 3.312 D.) 2.985 E.) 2.758←Letter corresponding to your answer to problem 5.Problem 6. (10 points if correct, -1 points if wrong). A cup of coffee initially at temperature T0= 172◦F is broughtinto a room at temperature A = 72◦F. The heat capacity of the room (compared to the coffee) is so large that we mayregard A as being constant. After 6 minutes the temperature of the coffee is 140◦F. What temperature will the coffeebe an additional 4 minutes later? Choose the closest value from the list below.A.) 120◦F B.) 125◦FC.) 130◦F D.) 135◦F E.) 140◦F←Letter corresponding to your answer to problem 6.Problem 7. (10 points if correct, -1 points if wrong). Find the general solution of the ordinary differential equationd3ydt3− 3d2ydt2− 4dydt+ 12y = 0.A.) y = c1e−3t+ c2e−2t+ c3e2tB.) y = c1e3t+ c2e2t+ c3te2tC.) y = c1e3t+ c2e2t+ c3e−2tD.) y = c1et+ c2tet+ c3t2etE.) None of the above.←Letter corresponding to your answer to problem 7.Problem 8. (10 points if correct, -1 points if wrong). Solve the ordinary differential equationd3ydt3−d2ydt2− 8dydt+ 12y = 0.A.) y = c1e2t+ c2te2t+ c3t2e2tB.) y = c1e3t+ c2e2t+ c3te2tC.) y = c1e−3t+ c2e−2t+ c3te−2tD.) y = c1e−3t+ c2e2t+ c3te2tE.) None of the above.←Letter corresponding to your answer to problem 8.This page is for scratch work. Use also the backs of the test pages.1 2 3 4 5 6 7 88 problems TOTAL:80