DEPARTMENT OF MATHEMATICS
MATH2001
Assignment 2
Summer 2015-2016
Submit your answers - along with this cover sheet - to your tutor at the end of the tutorial on
23/12/2015.
Note that you may nd some of these problems challenging. Attendance at tutorials i
DEPARTMENT OF MATHEMATICS
Revision solutions
(1) For this example we use partial fractions. We can write
1
A
B
=
+
(1 x)(2 + x)
1x 2+x
1 = A(2 + x) + B(1 x)
2A + B = 1 & A B = 0
1
A=B =
3Z
Z
Z
dx
1
dx
1
dx
=
+
(1 x)(2 + x)
3
1x 3
2+x
1
1
= ln |1 x| + l
DEPARTMENT OF MATHEMATICS
MATH2000/2001/7000
First Order ODEs solutions.
(1) (a)
The ODE is separable:
The equation can be written in the form
0
y =x
which is separable. Hence we have
Z
1 y2
y
y
dy =
1 y2
Z
x dx.
For the left hand side, substitute u = y 2
DEPARTMENT OF MATHEMATICS
MATH2000/MATH2001/MATH7000
Z
(1) For
Hyperbolic functions solutions
xdx
, set u = x2 du = 2xdx
1 + x4
Z
Z
x dx
1
du
=
4
2
1+x
1 + u2
Now set u = sinh t du = cosh tdt. Now the integral is:
Z
1
cosh tdt
p
2
1 + sinh2 t
Z
1
=
dt
2
a