Reference no: EM13983252

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.