Arizona State University Math 274 Differential Equations Test 1
100 Points Maximum: 75 Minutes Test: Graphing calculator not allowed for some problems: No Rubber-Necking: Show work: Graded test will be returned on 2/19/02.
Dr. S. Takahashi
Name:______
Your Predicted Score:______
Your Actual Score:______
This is a fun test. You should enjoy it.
1. Classify the following differential equations: You need to state whether each equation is:
(ア) Ordinary or Partial Differential Equation, (イ) Linear or Nonlinear equation, and state
(ウ) The Order of the equation. ( 12 points )
(a)
(b)
(c) ( This is the Laplace equation. It is and has been a hot item.)
(d) ( This is called the KP equation, becoming a hot item.)
2. Consider the following differential equation. Show work. ( 20 points )
a) Come up with all constant solutions.
b) Sketch the direction field for . Give sufficient data on your sketch. No Calculators.
c) Determine the end behavior of solutions. If end behavior is the function of the initial condition, , then describe this dependency.
d) Sketch a graph of the solution with the initial condition
3. Draw three isoclines with the direction field for the following differential equation. Also draw one integral curve of your choice. Give sufficient data. No calculators. ( 9 points )
4. Solve the following differential equation. ( 12 points )
5. Solve the following IVP. ( 12 points )
,
6. A tank initially contains 10 gallons of pure water. Brine with concentration of 2 lb/gal is poured into the tank at the rate of gal/min. The uniform solution is pumped out at the rate of 1 gal/min.
( lb/gal pounds per gallon, gal/min gallons per minute ) ( 15 points )
(ア) Calculate the concentration for the tank four minutes after you started to add brine to the tank.
(1) Suppose the rate coming in and the rate going out are identically the same (in gal/min), what can you conclude about the limiting concentration? Support your argument fully without solving the differential equation.
7. Come up with a suitable change of variable for the following differential equation so that the resultant
equation becomes a linear equation. Do not solve the equation. ( 5 points )
8. An object with mass 1 kg is shot upward with initial velocity of 40 meters/sec from the ground. Air resistance of is present, where is the velocity of the object measured in meters/sec. Calculate the time when the object reaches the maximum vertical distance above the ground. (15 pts.)