Math 182                                                PRACTICE EXAM 3

11/21/96

1. a) Find the general solution of the differential equation y'' - y' = 0. (Hint: let v = y' ).

b) Calculate the solution of the initial value problem y'' - y' = 0, y(0) = 0, y'(0) = 1.

2. Solve the following initial value problems:

a) dx/dt = 5 - 2 x, x(1) = -1

b) dy/dx = 3 xy cos(x²), y(0) =

3. a) Given the IVP dy/dx = x² + y², y(0) = 0 use Euler's method, with n = 3, to approximate the value of y(0.3). Keep all the decimals! Please, specify clearly all the pairs of numbers (x_i ,y_i) for 0 i 3.

b) Two slope fields are given below. Which one corresponds to dy/dx = x² + y² ? Justify your choice! Draw the solution that goes through the origin, and give a rough approximation of y(1.5).

4.    A substance, whose temperature is 45 degrees (Celsius), is put inside an oven maintained at the constant temperature of 120 degrees. After 2 minutes the temperature of the substance is 60 degrees. What will its temperature be seven minutes after the first measurement? Accept Newton's Law of Heating as your model.

5.   A reactor (volume = 10 liters ) is filled with a water solution containing 5 grams of salt.   We wish to dilute this solution. For this purpose, two valves, one at the top and the other at the bottom, are opened simultaneously letting in a solution of salt and water with a concentration of 0.1 grs./lt. Both valves work at a constant flow of 0.8 lt/min. For how long must the system operate in order to have 2 grams of salt inside the reactor?