1. Differential Equations
think about how the solution to a differential equation compares to solutions of other algebraic
equations you have solved in the past. How are solutions to differential equations similar or
different to what you have done before? Do some research on the applications of differential
equations, specifically the logistic equation. Where is the logistic equation most often used?
Answer. The solution of an algebraic equation represents one or more points on a 2dimensional coordinates plane or ordered pairs e.g. (2,5), (3,0) etc. while the solution of a
differential equation represents an expression or relation between dependent variable and
independent variable(s), e.g. 𝑦 = 3𝑥 + (1 −
The relation between the dependent and independent variable can be linear e.g. y = mx+b
or non-linear e.g. 𝑦 = 2𝑥 2 + 5𝑥 − 10.
Differential equations are extensively used in the advanced studies of physical science,
medical science, social science, economics etc. The logistic equation has applications to
those problems where the rate of change in the dependent variable at given value of the
independent variable is directly proportional to the dependent variable and the dependent
variable is under a constraint.
For example, if dependent variable is N and independent variable is t then logistic
= 𝑟𝑁 (1 − 𝑘 ) where k is the parameter or maximum value that N can
take. The logistic function is mostly used in the estimation of population of a nation or
2. Applications of Integration
sections that were covered focused on applications of integration. Of the applications that we
studied, which did you find the most interesting, and why? Do some of your own research to
find additional applications of integration. Compare what you found during your research to
what was covered in the book.
Answer. Integration is used to calculate the area between two curves, volume generated
by rotation of a curve about a straight line, to sum up a series and in solving many
scientific and economics problems. The application of integration to scientific problems
always impresses me. It is fascinated how complex and tedious problems are almost
correctly solved by the integration. In some cases, we derive relation between dependent
variable and independent variable(s). For example, the rate of change of velocity of a
moving body is proportional to the net force acting on the body.
This relation can be written ...
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