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

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step-by-step explanation: