Learning Outcome 1
Analyse engineering problems and formulate mathematical model using first order differential equations
1) The rate of cooling of a body is given by the equation:
dθ/dt = kθ
Where k is a constant.
If θ = 60°C when t = 4 minutes and
If θ = 30°C when t = 7 minutes
Determine the time taken for θ to fall to 20°C
Determine the equation of the curve which satisfies the equation:
xy dy/dx = x2 -1
which passes through the point (1,5).
Learning Outcome 2
Solve first order differential equations using analytical and numerical methods.
2) Find the solution of the following differential equations:
i. x.dy/dx - 4x = -y
ii. dy/dt + t = 5y
Use the Euler method to solve the differential equation with the conditions given:
dy/dx + 1 = -y/x
Initial conditions are:
y = 1 at x = 2
For a range of x = 2.0 to x = 2.5 with intervals of 0.1
Obtain a numerical solution of the differential equation and draw the graph of the solution.
Learning Outcome 3
Analyse engineering problems and formulate mathematical model using second order differential equations.
3) The displacement s of a body in a damped mechanical system, with no external forces, satisfies the following differential equation:
2(d2s)/(dt2) + 6ds/dt + 4.5s = 0
Where t represents time.
If initially, when t = 0, s = 0 and ds/dt = 4,
Solve the differential equation for s in terms of t.
Learning Outcome 4
4) Solve second order homogeneous and non-homogenous differential equations.
a. Find the general solutions of the following equations:
i. d2y/dx2 + dy/dt -2y = 0
ii. d2y/dx2 + 9y = 0
b. Find the general solutions of the following equation which satisfy the given initial conditions.
d2q/dt2 +dq/dt +q = t2 -1
q(0)=0
dq/dt =0, t=0
Learning Outcome 5
Apply first and second order differential equations to the solution of engineering situations
5) The velocity v of a rocket attempting to escape from the earth's gravitational field is given by:
vdv/dr = -gR2/r2
Where:
r is its distance from the centre of the earth and
R is the mean radius of the earth
Find a formula for V(r) and determine the minimum launch velocity V0 in order that the rocket escapes.