All problems in the tasks must be completed correctly with sufficient detail
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.
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
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
Solve second order homogeneous and non- homogenous differential equations.
6. Find general solutions of the following equations:
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
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:
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.