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MTH401 Differential Equations Assignment 2 Solution Fall 2012

Question#1                                                                                            Marks 10

 

(i) State the reason why the following differential equation is not linear?

(1)

(ii) Convert the differential equation in (i) into linear form.              (4)

 

(iii) Solve the following initial value problem:

(5)

Question#2                                                                                             Marks 10

 

(i) Find the general solution of the following differential equation:           (5)

(ii) Find the orthogonal trajectories of the general solution in (i).               (5)

Solution: 

Question#3                                                                                            Marks 10

 

(i) Construct a population dynamics model (exponential model) by considering the following assumptions:

  • measures the population of a species at any time.
  • is a differentiable, hence continuous, function of time.
  • indicates the time rate of change of the population of a species.
  • The rate of change of the population of a species is proportional to the existing population of a species.
  • represents the proportionality constant.

                                                                                                                                         (3)

(ii) Determine the solution of the model constructed in (i) subject to the condition    (is the initial population at).

(3)

(iii) If after two years the population of a country has doubled, and after three years the population is 20,000, then estimate the number of people initially living in the country using the model constructed in (i).

Solution:

Q3

Part i:    dN/dt = kN

part ii: above equation is linear as well a separable.

part iii : The rate of change of the population of a species is proportional to the existing population of a species.

PART IV: N( t) Measures the population of a species at any time t.

part v: k  greater than O for growth and K less than 0 for decay.

part vi:  N(2)= 7071

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