{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Exam 1 - Fall 2003

# Exam 1 - Fall 2003 - Brine containing ½ pound of salt per...

This preview shows page 1. Sign up to view the full content.

Differential Equations Exam #1 September 18, 2003 Name ________________________________ Show all your work neatly and in numerical order on notebook paper. DO NOT WRITE ON THE BACKS OF YOUR PAGES. 1. Given 0 = + x x : (a) Show that t c t c x sin cos 2 1 + = is a two-parameter family of solutions for the equation. (b) Find a particular solution satisfying the initial conditions ( 29 1 2 / , 0 ) 2 / ( = = π π x x . 2. Solve: ( 29 dy x y dx y 2 2 sec 1 = + 3. Solve: ( 29 1 1 2 = + + y y x 4. Solve: ( 29 x y dx dy x ln 1 = + + subject to the initial condition 10 ) 1 ( = y 5. Solve the exact equation: ( 29 ( 29 0 8 4 4 5 3 = - + + dy y x dx y x 6. Solve the Bernoulli equation: 2 y e y dx dy x = - 7. The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t. After 3 hours it is observed that there are 400 bacteria present. After 10 hours there are 2000 bacteria present. What was the initial number of bacteria? You must start by writing a differential equation and solving it. 8. A large tank is partially filled with 100 gallons of fluid in which 10 pounds of salt is dissolved.
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Brine containing ½ pound of salt per gallon is pumped into the tank at a rate of 6 gal/min. The well-mixed solution is then pumped out at a slower rate of 4 gal/min. Find the number of pounds of salt in the tank after 30 minutes. 9. A tank in the form of a right-circular cone standing on end, vertex down, is leaking water through a circular hole in its bottom. Suppose the tank is 20 feet high and has radius 8 feet and the circular hole has radius 2 inches. The differential equation governing the height h of water leaking from the tank is 2 / 3 6 5 h dt dh-= . In this model, friction and contraction of the water at the hole were taken into account with 6 . = c and 2 / 32 s ft g = . If the tank is initially full, how long will it take the tank to empty?...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online