All problems in the tasks must be completed correctly with sufficient detail

Task 1

Analyse engineering problems and formulate mathematical model using first order differential equations

1. Heated object is allowed to cool in the room temperature that has the constant temperature of T₀:

a. Analyse cooling process

b. Formulate mathematical model for the cooling process.

2. At time   t= 0 water begins to leak from the tank of constant cross-sectional area A.  Rate of outflow is proportional to h, depth of water in a tank at time t.  prepare constant of proportion kA where k is constant.

a. Analyse tank leaking process.

b. Formulate mathematical model for leaking process.

prepare conclusions based on the formulated mathematical model for the leaking process.

Task 2

Solve first order differential equations using analytical and numerical methods.

3. Find solution of the following equations:

a.   di/dt  =t(2-5i), i(0)=10

b.     t²(dq/dt) +tq=2

4. Utilize the Euler method with step size shown to advance four steps from given initial condition with given differential equation

dv/dt=2t+v,v(0) =1; h=0.1

Task 3

Analyse engineering problems and formulate mathematical model using second order differential equations.

5. For simple model of a shock observer shown in figure below:

a. Analyse the model, vertical motion of the mass.

b. Formulate the mathematical model of given model

Task 4

Solve second order homogeneous and non- homogenous differential equations.

6. Find general solutions of the following equations:

i. d²y/dx²+dy/dx-2y=0

ii. d²y/dx² + 9y=0

7. Find general solutions of the following equation that satisfy given initial conditions.

d²q/dt² + dq/dt+q= t²-1

q(0)=0 ,  dq/dt=0,   t=0

Task 5

Apply first and second order differential equations to solution of engineering situations

8. Velocity v of the rocket attempting to escape from the earth’s gravitational field is given by:

v(dv/dr)= -g(R²/r²)

Where:

r is its distance from the centre of the earth and

R is a mean radius of the earth

Determine the formula for V(r) and find the minimum launch velocity V0 in order that the rocket escapes.