CK12 Calculus - Single Variable

Resolvingthe partialfraction giveslinearequationsA+B= 2 andA-B= 0.

So .Integratingbothsides,wehave

or with. Then , i.e.
whereD> 0.

Therefore,the solutionhas form whereD> 0.
Exercise

1. Solvethe differentialequation whichsatisfiesthe conditiony(e) = 0.


2. Solvethe differentialequation.


3. Solvethe differentialequation.
   Exponentialand LogisticGrowth
   In somemodel,the populationgrowsat a rate proportionalto the currentpopulationwithoutrestrictions.

The populationis givenby the differentialequation , wherek> 0 are growth.In a refinedmodel,

the rate of growthis adjustedby anotherfactor whereKis thecarriercapacity.This is closeto
1 whenPis smallcomparedwithKbut closeto 0 whenPis closetoK.
Both differentialequationsare separableand couldbe solvedas in last section.The solutionsare respectively:

and with.

Example 1 (ExponentialGrowth)The populationof a groupof immigrantincreasedfrom to
from the end of first year to the end of secondyear they cameto an island.Assumingan exponentialgrowth
modelon the population,estimatethe size of the groupof initialimmigrants.

Solution.The populationof the groupis givenby wherethe initialpopulationand relativegrowth
rate are to be determined.

At (year), , so.

At (year), , so.

Dividingboth sidesof the secondequationby the first, we have.