Differential Equations Practice Exam 3



Differential Equations Practice Exam 3

1. (Exam 2 Fall 1996-Problem 1) Given:

(a) [pic]

(b) [pic]

For each differential equation,

(i) find two linearly independent real solutions. (You need not verify that the solutions are linearly independent.)

(ii) solve the initial value problem [pic].

2. (Exam 2 Fall 1996 – Problem 2) Given [pic].

a) Find the general solution of the associated homogeneous equation.

b) Find the general solution to the original non-homogeneous equation.

3. (Exam 2 Fall 1996- Problem 5)

a) Determine if the method of undetermined coefficients is applicable for each of the differential equations. Answer YES if applicable, NO if not.

b) If the answer is YES, find suitable forms of the particular solution [pic].

i. [pic]

ii. [pic]

iii. [pic]

iv. [pic]

v. [pic]

4. (Exam 2 Fall 1996-Problem 6) A mass weighing 8 lb stretches a spring 3 in. to the rest position. The mass is also attached to a damper with coefficient [pic].

a) Evaluate the spring constant and write down the equation of the motion of the mass without exterior force.

b) For which values of [pic], do all solutions have infinitely many zeros?

c) For which values of [pic], do all non-trivial solutions (a solution is non-trivial if it is not constant zero) have only finitely many zeros? How many zeros can such a solution have?

5. (Exam 3 Fall 1995- Problem 2) Given [pic].

a) Let [pic], [pic], [pic]to convert the equation into a system of 3 coupled first order equations.

b) Write the system in (a) in matrix-vector form, [pic].

c) Find the characteristic polynomial for A, [pic]

d) Find the eigenvalues of A.

6. (Exam 3 Fall 1997- Problem 4) Consider the following matrix [pic]

a) Compute the eigenvalues [pic] of A.

b) Computer the eigenvectors of A associated with the eigenvalues [pic].

7. (Exam 3 Fall 1997- Problem 3). For the system of differential equations

[pic].

a) Find the general solution of the equation.

b) Sketch the phase portrait of the system.

c) Find the solution for the case [pic]

d) Find a solution for which [pic] as [pic].

8. (Exam 3 Fall 2001- Problem 2) Consider the system of ODEs [pic].

a) If A is a defective matrix, with a double eigenvalue [pic] that has only one eigenvector [pic].

(i) Show that [pic] is a solution of [pic].

(ii) Derive the condition that vector [pic] has to satisfy such that [pic]is also a solution of [pic].

b) Given the (non-defective) matrix [pic], find the general solution of [pic]. Express your final result in terms of real functions.

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