ENGR 311 TRANSFORM CALCULUS AND PARTIAL DIFFERENTIAL EQUATIONS

CONCORDIA UNIVERSITY
FACULTY OF ENGINEERING AND COMPUTER SCIENCE
DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING

ENGR 311 TRANSFORM CALCULUS AND PARTIAL DIFFERENTIAL EQUATIONS
Mid Term Examination (May 28, 2026)


Total Marks: 30 Time: 18:30 – 20:30
Individual Work – closed book/notes test;
Materials allowed: approved calculator;
You must show all the steps in the calculations(s) to get the full mark assigned to the question.


1 Use the Laplace transform to solve the following differential equations (5)
.
(a y '' −3 y' +2 y=e−4t , y (0)=5, y' ( 0 )=5
) Laplace transform of DE yields

L { } {}
d2
dt2
−3 L
dy
dx
+2 L { y }=L {e −4 t }
1
s Y ( s)−sy (0)− y (0)−3 [sY (s )− y (0) ] +2 sY ( s )=
2 '

s+ 4
Putting the initial values in the equation and solving for Y(s)
s
1
(¿¿ 2−3 s+2)Y ( s)=5 s−10+
s +4
¿
s
(¿¿ 2−3 s +2)( s+ 4)
Y (s )= 2 5 s − 2 10 +1
s −32s +2 s −3 s+ 2 ¿
5 s +10 s−39
Y (s )=
( s−1 ) ( s−2 ) ( s +4 )
24 1 1 1 1 1
Y (s )=¿ + +
5 s−1 6 s−2 30 s+ 4
Taking inverse Laplace transform
24 1 1
y (t )= et+ e2t+ e−4 t
5 6 30

(b y + 2 y + 2 y =∂(t - u ) (5)
) Laplace transform of DE yields
[ s 2 Y (s)−sy (0)− y' (0 )]+2 [ sY (s )− y (0)]+2 Y ( s)=e−πs
Putting the initial values in the equation and solving for Y(s)
[ s 2 Y ( s)−0−0 ] +2 [ sY (s )−0 ]+ 2Y ( s)=e−πs
[ s 2 +2 s+ 2 ] Y ( s)=e−πs


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, e−πs
Y ( s)=
( s+1)2+ 1
Taking inverse Laplace transform
y (t )=e−(t−π) sin(t −π )u (t −π )

2 (a Express the following function in terms of unit step functions and (2.5
. ) evaluate )
L{f (t)}.
f (t)
100


t
5

{
f (t )= 20 t , 0 ≤t ∈5
0, t ≥ 5
f (t )=20 t −20 t u(t −5)


(b
20
( ) 1 5
F ( s)= s2 −20 e − 5s s2 + s
Find the inverse Laplace transform of: (5)
)
2 - 3se- s + 4e- s
F (s) =
s(s +1)

2
(
3 s−4
F ( s)= s( s+1) − s( s+1) e−s )
2 2 4 7
¿ s − s +1+ s − s+1 e−s( )
f ( t )=2−2e−t+ 4 u ( t −1)−7 e−(t−1)u( t−1)
3 Find the Laplace transform of the given periodic function. (2.5
)

f (t)
1
t
a 2a 3a 4a
–1

The function f(t) has amplitude 1 and interval 2a. On interval 0 ≤t ∈2 a , f(t) can be
defined by

{
f (t )= 1, 0 ≤ t ∈ a
−1, a≤ t ∈2 a

Using the theorem of a periodic function, the Laplace transform of the function can be
written as


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