MATH 2065 Midterm: MATH 2065 LSU f08Exam 1
Name: Exam 1
Instructions. Answer each of the questions on your own paper. Be sure to show your
work so that partial credit can be adequately assessed. Credit will not be given for answers
(even correct ones) without supporting work. Put your name on each page of your paper. A
short table of integrals is included on Page 2.
1. [18 Points] Solve the initial value problem: y′= 4t3y2,y(0) = 1/4
2. [18 Points] Find the general solution of: y′−3y= 5e3t+et
3. [18 Points] Solve the initial value problem: ty′+ 4y=t,y(1) = −1
4. [18 Points] Solve the initial value problem: (4t+ 4y+ 3) + (4t−6y−2)y′= 0,
y(2) = 1
5. [12 Points] Apply Picard’s method to compute the approximations y1(t), y2(t), and
y3(t) to the solution of the initial value problem
y′= 2y−t, y(0) = 0.
6. [16 Points] A 2000 gallon tank is initially full of brine which contains 100 pounds of
salt. A solution containing 3.0 pounds of salt per gallon enters the tank at a flow rate
of 4 gallons per minute. A drain is opened at the bottom of the tank through which
the well stirred solution leaves the tank at the same flow rate of 4 gallons per minute.
Let y(t) denote the amount of salt (in pounds) which is in the tank at time t.
(a) What is y(0)? That is, how much salt is in the tank at time t= 0?
(b) Find the amount y(t) of salt in the tank for all times t.
(c) How much salt is in the tank after 30 minutes?
(d) What is limt→∞ y(t)?
Math 2065 Section 1 September 29, 2008 1
Document Summary
Answer each of the questions on your own paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without supporting work. Put your name on each page of your paper. A solution containing 3. 0 pounds of salt per gallon enters the tank at a ow rate of 4 gallons per minute. A drain is opened at the bottom of the tank through which the well stirred solution leaves the tank at the same ow rate of 4 gallons per minute. Some integral formulas: r xn dx = 1 n+1xn+1 + c (if n 6= 1, r. 1 x dx = ln |x| + c dx = 1 b ln |a + bx| + c (b 6= 0) 4. 1 a2 + x2 dx = tan 1 x a. 1: r, r x(a + bx) dx =
