can you make sure to solve all the questions accurately as it is a very important a1. (a) By making the substitution u = e
x and then using partial fractions, evaluate Z
dx
e
x + 1.
(b) By making a suitable substitution, evaluate Z
cos θ dθ
sin3
θ + sin θ.
2. A differential equation of the form
dy
dx + p(x)y = q(x)y
n
is called a Bernoulli differential equation.
(a) Show that if y is the solution of the above Bernoulli differential equation and u = y
1-n, then u satisfies
the linear differential equation
du
dx + (1 - n)p(x)u = (1 - n)q(x).
(b) Hence solve the differential equation
x
2
y
0 + 2xy = y
3
.for this semesterSolve the differential equation
y
00 - 4y
0 + 13y = 6e
2x
cos 3x
subject to the initial conditions y(0) = 3 and y
0
(0) = -8.
4. Solve the system of differential equations
x
0
y
0
-
-11 15
-30 31 x
y
= e
t
18
30