Consider the differential equation y'+\lambda y=e^{-t}, when \lambda is some constant.
(a) find all values of \lambda such that all solutions tend to zero as t \rightarrow infinity .
(b) at least one solution goes to zero as t \rightarrow infinity .

Part a) value of such that all the solutions tend to zero equals 1.

Part b)

For a particular solution to tend to 0 will depend on the boundary conditions.

Step-by-step explanation:

The given differential equation is

This is a linear differential equation of first order of form whose solution is given by

Applying values we get

here are arbitrary constants

part 1)

For all the function to approach 0 as t approaches infinity we have

Part b)

For a particular solution to tend to 0 will depend on the boundary conditions as are arbitrary constants

introduction, methodology, results, discussion

step-by-step explanation:

an observational study is one which consists of actual observation to record data and then analyse the data.

there are a number of factors that should be analyzed before starting an observational study.

the main four factors are:

i) introduction:   this should be analysed to check whether the purpose of the study is correctly established and the data recorded are appropriate for the study

methodology:   the method of collecting the data, such as whether randomly selected, and whether sample size matches with the population to be studied, whether questions asked are easy to answer by common men, etc.

results: then results such as summary of data with comments should be studied thoroughly

discussion:   the last is the discussion, this can be either by management people or persons who are all interested , etc.

whatvare the choices

step-by-step explanation: