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 T_{0}:
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. t^{2}(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. d^{2}y/dx^{2}+dy/dx-2y=0
ii. d^{2}y/dx^{2} + 9y=0
7. Find general solutions of the following equation that satisfy given initial conditions.
d^{2}q/dt^{2} + dq/dt+q= t^{2}-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(R^{2}/r^{2})
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.