MATH 267 Section P Fall 2015 PRACTICE FINAL EXAM

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MATH 267 Section P Fall 2015 PRACTICE FINAL EXAM

1. Find the general solution of y

0

= 3 y

4

8 xy

4

2. Solve the initial value problem y

00

+ 4 y

0

+ 13 y = 0 y (0) = 2 y

0

(0) = 1

3. Use the substitution z = ln y to find the general solution of the equation y

0

+ e x y = y ln y

4. Find the general solution of y

00

+ 2 y

0

+ y = t

2

− cos 4 t

5. Use the variation of parameters method to find a solution of y

00

+ 4 y = sec 2 t

6. Let

1

A =

6

0

(a) Find the eigenvalues and eigenvectors of A .

(b) Find the general solution of x

0

= A x .

0

1

4

0

1

2

7. Let

A =

1

5

1

3

(a) Find a fundamental matrix for x

0

= A x .

(b) Find the solution of x

0

= A x satisfying the initial condition x (0) =

1

1

8. The system x

0 y

0

=

= y y

− x has (0 , 0) as its only equilibrium point. Classify this equilibrium point by stability and type, and sketch the phase portrait near (0 , 0).

9. Let f ( t ) =

(

2

0

(a) Find the Laplace transform of f .

(b) Use the Laplace transform method to solve y

00

9 y = f ( t )

0 < t < 3 t > 3 y (0) = 0 y

0

(0) = 1

10. We seek a solution of y

00

+ x

2 y

0

3 y = 0 in the power series form y ( x ) =

P

∞ n =0 a n x n

.

(a) What is the recurrence relation for the coefficients?

(b) If y (0) = 2 and y

0

(0) = 0 find a n for n = 0 , 1 , 2 , 3.

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