UNIVERSITY OF DUBLIN
MA2E21
TRINITY COLLEGE
Faculty of Science
school of mathematics
SF Engineering
Trinity Term 2000
Course 2E2
Tuesday, June 6
Luce Hall
14.00 — 17.00
Dr. P. Tran Ngoc Bich
Answer ALL questions
1. Solve the Initial Value Problem
y 00 + 4 y 0 + 3 y = et + 2 δ(t)
where
y(0) = y 0 (0) = 0 .
2. Using the Second Shifting Theorem, find the Laplace transform of the function
f (t) = (t − 1) H(t − 1) + H(t − 1) ,
where H is the Heaviside function. Solve the Initial Value Problem:
y 00 − y = f (t)
where
y(0) = y 0 (0) = 0 .
Calculate the first derivative of the solution y(t).
3. By means of the Z-transform, solve the difference equation
un+2 − 3 un+1 + 2 un = 0
where
u0 = 1 , u1 = −4 .
4. Using the matrix form of the Laplace transform, solve the system of differential
equations
x01 = 2 x1 − x2 + 3 x3 + (1 − t) ,
x02 = −x1 + x2 − x3 + (1 + t) ,
x03 = x2 − x3 + t .
for the initial values: x1 (0) = x2 (0) = x3 (0) = 1.
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MA2E21
5. Using the Power Series method, solve the differential equation
y 00 − x y = 0
where
y(0) = 1 , y 0 (0) = 0 .
6. Show that x0 = 1 is a regular singular point of the differential equation
(1 − x2 ) y 00 − 2x y 0 + 2 y = 0 .
Find the indicial equation of this differential equation. Deduce the general form of
the solution (do not compute explicitly the solution).
~ · F~ and ∇
~ × F~ for
7. Compute ∇
F~ = sin(x − z)~ı + y 2 z ~ + z 2 y ~k .
8. State Green’s theorem in the plane. Evaluate
Z 2
2x cos(2y) dx − 2x sin(2y) dy
C
around the circle C of centre the origin and radius one.
c UNIVERSITY OF DUBLIN 2000