[answered] Math 342, Final Exam due on December 7 no later than 4:00 p

Math 342, Final Exam

due on December 7 no later than 4:00 pm

1. Consider the equation

x2 y 00 + xy 0 + (x2 ? 1)y = 0.

(1)

(2)

(3)

(4)

(5) Show that x = 0 is a regular singular point.

Find the roots of the indicial equation.

P

(?1)n x2n

Show that one solution for x &gt; 0 is of the form y1 (x) = x2 ?

n=0 (n+1)!n!22n .

Show that y1 converges for all x real.

P

n

Prove that it is impossible to find a solution of the form x1 ?

n=0 bn x , x &gt; 0. 2. Consider the matrix ? 1 0

A = 0 ? 1 .

0 0 ?

(1) Compute A2 , A3 , and A4 .

(2) Use an inductive argument to show that n

? n?n?1 [n(n ? 1)/2]?n?2

.

?n

n?n?1

An = 0

n

0

0

?

(3) Determine eAt .

3. Consider the initial value problem

y 0 = 2y ? 3t, y(0) = 1.

Use the fourth-order Adams-Moulton method with constant step size h = 0.01 to

compute an approximate solution in the interval [0, 1]. 4. Consider the system of equations

(

x0 = y + ?x ? x(x2 + y 2 )

y 0 = ?y ? x ? y(x2 + y 2 ).

(1) Show that (0, 0) is the only critical point.

(2) Find the linear system that approximates the given system near (0, 0). Determine

the type and stability of (0, 0) for the original system relative to the parameter ?.

(3) Show that for ? &gt; 0 the original system has a single periodic orbit.

(4) Sketch the phase plane portrait for ? &gt; 0 and explain how it varies with ?.

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