7/25/2019 UMD ECE Qualify Basic Maths 2015 Fall

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Basic Math SolutionsFall 2015

(1) Find the general and particular solutions of the differential equation

)sin(6222

2

3

3

xxy

dx

dy

dx

yd

dx

yd

SOLUTION

Auxiliary equation is 022 23

mmm with solution

21121 2 mmmmmm so general solution is

xxx CeBeAey 2

Particular solution has to be of the form )cos()sin( xdxcbaxy

Substitution into original differential equation gives

)cos(sin23101

21

xxxy

(2) Evaluate the line integral C dsxyI 2

Along the curve the )sin(),cos( tytx from t=0 to t=/4

SOLUTION

Along the curve dtdt

dy

dt

dxds

22

where )(),( tgytfx

Therefore 2

1)2sin()cos()sin()sin()cos(2

4/

0

4/

0

22

dttdtttttI

(3) Find the eigenvalues and normalized eigenvectors of the matrix

332

021

001

A

SOLUTION

For this lower triangular matrix the eigenvalues are on the diagonal and are =1, 2, 3

7/25/2019 UMD ECE Qualify Basic Maths 2015 Fall

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To find eigenvectors solve

z

y

x

z

y

x

332

021

001

for each eigenvalue

The normalized eigenvectors are

2

1

11

3

2,

3

10

10

1and

1

00

corresponding to the three eigenvalues

1,2,and 3.