Mth 256 Test 1 Name: ID:Bent Petersen 256w2002z-test1.tex Feb 6, 2002 Time: 50 minutes.You may use a notesheet, prepared in advance, and no larger than 8.5 × 11 inches in size. You are expected to have a scientificcalculator, and you may use it. Please note log(x) means the natural logarithm of x on this test.This test is multiple-choice. Work carefully. Try to avoid errors and try to avoid being misled by the offered answers.There are 8 problems for a total of 160 points.Problem 1. (20 points if correct, 0 points if wrong). Find the general solution of the differential equationxdydx=1 + x3 1 + y2.A.) y = tanx +x33+ CB.) y = tanx44+ log(x) + CC.) y = logx33+ log(x) + CD.) y = tan (arctan(x) + C) E.) None of the foregoing.←Letter corresponding to your answer to problem 1.Problem 2. (20 points if correct, 0 points if wrong). Solve the initial value problemdydx= e−x−y, y(0) = log 3.A.) y = log (1 + ex) B.) y = log (3ex− 1) − xC.) y = log (4ex− 1) − x D.) y = log (2 + x) E.) None of the foregoing.←Letter corresponding to your answer to problem 2.Problem 3. (20 points if correct, 0 points if wrong). Solve the differential equationxdydx+ y = 3x2.A.) y = x2+ C B.) y = x2+ C/xC.) y = x3+ C/x D.) y = x3+ C E.) None of the foregoing.←Letter corresponding to your answer to problem 3.Problem 4. (20 points if correct, 0 points if wrong). Solve the exact ordinary differential equation2xy + y2− y + 3x2+ 1 + (x2+ 2xy − x − 4y + 2)dydx= 0A.) x2y + xy2− xy + x3− 2y2+ x + 2y = CB.) x2y + 2xy2− xy + x3− 2y2+ x + 2y = CC.) x2y + xy2− 2xy + x3− 2y2+ x + 2y = CD.) x2y + xy2− xy + 2x3− 2y2+ x + 2y = CE.) None of the foregoing.←Letter corresponding to your answer to problem 4.Problem 5. (20 points if correct, 0 points if wrong). The ordinary differential equationy(x + y + 1) + (x + 2y)dydx= 0has an integrating factor depending only on x. Find such an integrating factor.A.) x2B.) log(x)C.) exD.) x E.) None of the foregoing.←Letter corresponding to your answer to problem 5.Problem 6. (20 points if correct, 0 points if wrong). If we substitute y = xv in the differential equationdydx=xy + y2+ x2x2we obtainA.)dvdx= v2+ 1 B.) xdvdx= v2+ 1C.) xdvdx= v2+ v + 1 D.) xdvdx= v2+ 2v + 1 E.) None of the foregoing.←Letter corresponding to your answer to problem 6.Problem 7. (20 points if correct, 0 points if wrong). A very large tank contains 40 L brine of concentration 2 g/L salt. Brineof concentration 1.2 g/L salt flows into the tank at 4 L/min and the well-mixed solution is pumped out at 2 L/min. Assuming thatthe tank does not overflow what is the concentration of salt in the brine in the tank after 10 min? (Choose the closest value.)A.) 1.555 g/L B.) 1.448 g/LC.) 1.327 g/L D.) 1.298 g/L E.) 1.000 g/L←Letter corresponding to your answer to problem 7.Problem 8. (20 points if correct, 0 points if wrong). Given the initial value problemdydx= x + y2, y(0) = 1estimate y(0.6) by using EULER’s method with step size h = 0.2. Choose the closest number from the list below.A.) 1.97560 B.) 2.07495C.) 2.64399 D.) 2.94948 E.) 3.11037←Letter corresponding to your answer to problem