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examfinalW09

# examfinalW09 - Name Section/Time of lecture Professor/GSI...

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Name: Section/Time of lecture: Professor/GSI: MATH 216 WINTER 2009 FINAL EXAM To get full score you need to carefully explain what you did. No calculators allowed. LAPLACE TRANSFORM TABLE IS ON THE SECOND PAGE 1

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Problem Points Score 1 30 2 20 3 26 4 30 + 2 2 TOTAL 108 TABLE OF LAPLACE TRANSFORMS L ( f ( t )) = F ( s ) L ( t n ) = n ! s n + 1 L ( sin at ) = a s 2 + a 2 L ( cos at ) = s s 2 + a 2 L ( e at ) = 1 s a L ( e at f ( t )) = F ( s a ) L ( u ( t a ) f ( t a )) = e as F ( s ) L ( δ a ( t )) = e as L ( t f ( t )) = F ( s ) L t 0 f ( τ ) d τ = F ( s ) / s L ( f ( n ) ) = s n F ( s ) s n 1 f ( 0 ) s n 2 f ( 0 ) · · · s f n 2 ( 0 ) f n 1 ( 0 ) L ( sin kt kt cos kt ) = 2 k 3 ( s 2 + k 2 ) 2 L ( t sin kt ) = 2 ks ( s 2 + k 2 ) 2 2
Problem 1. a-10pt Solve t dy dt + 2 y = sin t t , y ( π ) = 1. 3

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b-10pt Find a (particular) solution for y ( 5/ t ) y + ( 5/ t 2 ) y = 8 t 3 , given that the functions y 1 ( t ) = t and y 2 ( t ) = t 5 solve the corresponding homogeneous differential equation. 4
c-10pt Find the general solution for y ( 4 ) 3 y 4 y = 0 5

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Problem 2. a-10pt Solve y 4 y + 3 y = 2 e t , y ( 0 ) = 3, y ( 0 ) = 6. 6
b-10pt Solve the following system of ordinary differential equations x = 5 x + 4 y y = 4 x + 5 y 7

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Problem 3. a-6pt Consider the following system dx / dt = x 2 y + 1 dy / dt = xy y 2 + 2. Given that ( 1, 2 ) is a critical point of the system, discuss its stability.

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examfinalW09 - Name Section/Time of lecture Professor/GSI...

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