examfinalW09 - Name: Section/Time of lecture:...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 Problem Points Score 1 30 2 20 3 26 4 30 + 2 2 TOTAL 108 TABLE OF LAPLACE TRANSFORMS L( f (t)) = F (s) n! L ( t n ) = n +1 s a L(sin at) = 2 s + a2 s L(cos at) = 2 s + a2 1 L(e at ) = s−a at L(e f (t)) = F (s − a) L(u(t − a) f (t − a)) = e−as F (s) L(δa (t)) = e−as L(−t f (t)) = F ￿ (s) ￿￿ t ￿ L f (τ )dτ = F (s)/s 0 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 ) 2k 3 ( s2 + k 2 )2 2ks L(t sin kt) = 2 + k 2 )2 (s L(sin kt − kt cos kt) = 2 Problem 1. a-10pt Solve t dy sin t + 2y = , y(π ) = 1. dt t 3 b-10pt Find a (particular) solution for y￿￿ − (5/t)y￿ + (5/t2 )y = 8t3 , given that the functions y1 (t) = t and y2 (t) = t5 solve the corresponding homogeneous differential equation. 4 c-10pt Find the general solution for y(4) − 3y￿￿ − 4y = 0 5 Problem 2. a-10pt Solve y￿￿ − 4y￿ + 3y = 2et , y(0) = 3, y￿ (0) = 6. 6 b-10pt Solve the following system of ordinary differential equations x ￿ = 5 x + 4y y ￿ = −4 x + 5y 7 Problem 3. a-6pt Consider the following system dx /dt = x2 − y + 1 dy/dt = xy − y2 + 2. Given that (1, 2) is a critical point of the system, discuss its stability. 8 b-6pt Let mx ￿￿ = −kx + β x3 be the equation for a spring, m = k = β = 1. Write this as a first order system and find the critical points ( x, y). x￿ = y y￿ = · · · 9 c-10pt Calculate the Laplace transform F (s) of the function f (t) when 0 if t < 2, f (t) = 4 if 2 ≤ t < 4 t if t > 4. 10 Problem 4. a-10pt Find the inverse Laplace transform of F (s) = 11 11 . s2 s +1 b-10pt Use Laplace transform to solve y￿￿ − 2y￿ + 2y = 0, y(0) = 0, y￿ (0) = 1. 12 c-10pt Solve the initial value problem y￿￿ + y = f (t), y(0) = y￿ (0) = 0 where f (t) = 13 ￿ 0, 1, t<2 . t≥2 ...
View Full Document

This note was uploaded on 01/30/2012 for the course MATH 216 taught by Professor Stenstones? during the Fall '07 term at University of Michigan.

Ask a homework question - tutors are online