Force, momentum, impulsive force, impulse,
Dirac delta function
Let P(t) and F(t) be a momentum
and force at time t respectively
It is know that dP (t )
= F (t )
dt
t2
Hence
P (t ) P (t ) =
F (t )dt
2
1
t1
i.e., change of momentum from t2 to t1
= F (t )dt

MA1506 TUTORIAL 10
Question 1
Wherever possible, diagonalize the matrices in question 3 of Tutorial 9. [That is, write
them in the form PDP1 , after nding P and D, where D is diagonal.] Using this, work
out their 4th powers.
[Answers: The rst one cannot b

ANSWERS TO MA1506 TUTORIAL 7
Question 1.
(a) We shall use the following s-Shifting property:
L(f (t) = F (s) L(ect f (t) = F (s c)
n!
2
2
n
L(t ) = 3 use L(t ) = n+1
s
s
2
2 3t
) = L(e3t t2 ) =
L(t e
(s + 3)3
(b) Here u denotes the Unit Step Function gi

ANSWERS TO MA1506 TUTORIAL 6
Question1.
B2
.
4s
1.5
From Tutorial 5 we know B = 1.5 and and N = 376, so N = B/s s = 376
2
B
4s = 141. This is the maximum number we can kill without causing extinction.
First compare 80 with
Setting E = 80,
B
1
=
2
B 2 4Es

MA1506 Tutorial 3 Solutions
(1a)
y"6 y '9 y 0
Set y e t
2 6 9 0 3
y ( A Bx )e 3 x y ' Be 3 x 3( A Bx )e 3 x
y (0) 1 A 1
y ' 0 1 B 3 A 1 B 2 y (1 2 x)e 3 x
(1b)
2 2 (1 4 2 ) 0 1 2i
y e x [ A cos 2x B sin 2x]
y ' y e x [2A sin 2x 2B cos 2x]
y (0) 2 A 2
y

MA1506
Tutorial 9
SOLUTIONS
Question 1. The thing to remember here is that there are six possible states in which
the cunning Miss Tan can find herself: she can have $0, $1, $2.up to $5. She can never
have more than that because Ah Huat wont allow it. If

MA1506 TUTORIAL 4 SOLUTIONS
Question 1
(i) x
= cosh(x). An equilibrium solution of an ODE is just a solution that is identically
constant. That is not possible here because the cosh function never vanishes. So there is
no equilibrium for this ODE.
(ii) x

MA1506 TUTORIAL 8 SOLUTIONS
Question 1
From the given hint, we see that we have to solve
d4 y
Mg
=
(x A),
dx4
EI
subject to the given boundary conditions. [Note that y(0) = y (0) = 0 since the pole is
horizontal at the point where it joins the wall.] Taki

ANSWERS TO MA1506 TUTORIAL 5
Question 1
If the weight per unit length is 2[1 (x/L)], then the total weight is obtained by
integrating this from 0 to L; the answer is L, which is indeed exactly the same weight as
Ah Huats balcony [which has constant weight

MA1506 TUTORIAL 9
Question 1
Billionaire engineer Tan Ah Lian believes that she can get even richer by gambling. To
this end, she goes to an Integrated Resort1 and plays the following game [along with
several other players]. The players and the croupier e

MA1506 TUTORIAL 4
Question 1
Find the equilibrium points for the following equations. Determine whether these equilibrium points are stable, and, if so, nd the approximate angular frequency of oscillation
around those equilibria. In the second example, us

MA1506 TUTORIAL 7
1. Find the Laplace transforms of the following functions [where u denotes
the unit step function and the answers are given in brackets]:
2
(a) t2 e3t .
[
]
(s + 3)3
1
2
(b) tu(t 2).
[e2s cfw_ 2 + ]
s
s
2. Find the inverse Laplace transf

Remarks of Tutorial 4
Q1
(a) Recall : A solution of a given ODE is said to be
an equilibrium solution (point) if it is
a CONSTANT solution.
If xE is an equilibrium pt of
x = f ( x)
then
Hence
xE = f ( xE )
and
xE = 0
f ( xE ) = 0
So finding equilibrium

Remarks on T1
d
= tan
Q4 Solve r
dr
is fixed
What will happen to the moth?
To answer this question, discuss
the solution of the above ODE
There are three cases:
>90
<90 =90
We will understand these three cases better
if we sketch the graph of the solutio

MA1506 TUTORIAL 3
1.
Solve the following dierential equations:
(a) y + 6y + 9y = 0,
y(0) = 1,
(b) y 2y + (1 + 4 2 )y = 0,
2.
(c) y y = 2xsin(x)
y (0) = 2(3 1)
(b) y 6y + 8y = x2 e3x
(d) y + 4y = sin2 (x)
Use the method of variation of parameters to nd par

MA1506 TUTORIAL 6
Question 1
In question 6 of Tutorial 5, let us assume that you are keeping the bugs not as a hobby,
but because you are developing a new insecticide. Suppose that you remove 80 bugs per
day from the bottle, and that all of these bugs die

Remarks on Tutorial 2
Q2 Find function y(x) such that
Derivative at pt t
Derivative at pt x
dy x dy
=
+ 1dt
y '( x)
dx T 0 dx =
2
( y '(t ) )
T
x
0
2
+ 1dt
i.e., solving the above integral equation
Need to change to solving differential equation
Let u(x)

MA1506 TUTORIAL 8
Question 1
Despite her vast wealth, Tan Ah Lian continues to live with her mother. The latter does
TALs laundry [of course], including a large heavy jacket. TALs mother hangs the jacket
out to dry on a bamboo pole inserted into a socket

MA1506 TUTORIAL 5
Question 1
Tan Ah Lian, billionaire engineer, has a business which was awarded a huge contract to
retro-t a large balcony onto every HDB apartment in Singapore. The contract xes the
physical properties and the total weight W of the build

Remarks on T7
Q5 see appendix
Q6
The force exerted by the rogue wave
in upward direction
T +
=
P =
F (t )dt F
T
over t=T to t=T+ is - P/
Q7 We can check that
When -T/2<t<T/2
T (t=
)
k =
T (t )
(t kT=)
is T-periodic
(t )
see appendix
k =
Use this result