Midterm

Differential Equations

Fall 2004, Hunsicker

 

Instructions:  You have three hours to complete this exam.  No books or notes or any other resources such as your classmates are permitted.  You may use a calculator for algebraic calculations or graphing functions, but you must do all other work by yourself, and show your work.  Except in the case of numerical calculations, indicate when and how you have used your calculator.  Unless you have made prior arrangements with me, you must take the exam in the exam room.  There are 100 points possible on this exam.  Points for each problem are given after the problem number.  Allot your time accordingly.

 

Definitions and Theorems

1)    (5 points)  Define bifurcation point.

2)    (15 points)  Define autonomous o.d.e, equilibrium solution, stable, semi-stable and unstable.  Explain what these ideas mean in terms of slope field diagrams using pictures.  Explain what these ideas mean in terms of applications using examples.

3)    (10 points) State the existence and uniqueness theorem for first order differential equations.

4)    (10 points)  State and prove the principle of superposition.

 

 

Calculations

5)    (10 points)  Solve the following equation: 

2t cosy + 3t 2y + (t 3 – t 2sin(y) –y) yÕ = 0.

 

6)    (10 points)  Solve the initial value problem:

yÓ – 3yÕ – 4y = 0

y(1) = 0

yÕ(1) = 2

 

7)    a)  (10 points) Use the Euler method with h=.5 to estimate the solution to the following IVP at t=2:

yÕ = (1+t)(1+y)

y(0) =2

 

b)    (10 points)  Now solve the equation and compare its value at t = 2 to your estimate.

 

Problems

8)    (10 points)  A tank contains 300 gallons of water and 100 pounds of pollutants.  Fresh water is pumped into the tank at 2 gal/min and a well-stirred mixture leaves the tank at the same rate.  How long does it take for the concentration of pollutants in the tank to decrease to 1/10 of its original value?

9)    a)  (5 points)  Calculate the Wronskian of the functions y1(t) = exp(-t2) and y2(t)=(1+t2) -1.

 

b)  (5 points)  Could y1 and y2 be two solutions to a second order linear homogeneous equation with coefficients p(t) and q(t) continuous on the interval

(-1,1)?  Why or why not?