The InitialValue Problem
LearningObjectives
- Find generalsolutionsof differentialequations
- Use initialconditionsto find particularsolutionsof differentialequations
Introduction
In the Lessonon IndefiniteIntegralsCalculuswe discussedhow findingantiderivativescan be thoughtof
as findingsolutionsto differentialequations: We now look to extendthis discussionby
lookingat how we can designateand find particularsolutionsto differentialequations.
Let’s recallthat a generaldifferentialequationwill havean infinitenumberof solutions.We will look at one
suchequationand see how we can imposeconditionsthat will specifyexactlyone particularsolution.
Example1:
Supposewe wish to solvethe followingequation:
Solution:
We can solvethe equationby integrationand we have
We note that thereare an infinitenumberof solutions.In someapplications,we wouldlike to designateexactly
one solution.In orderto do so, we needto imposea conditionon the function We can do this by speci-
fying the valueof for a particularvalueof In this problem,supposethat add the conditionthat
This will specifyexactlyone valueof and thus one particularsolutionof the originalequation:
Substituting intoourgeneralsolution gives
or Hencethe solution is the particularsolution of the
originalequation satisfyingtheinitialcondition
We now can think of otherproblemsthat can be statedas differentialequationswith initialconditions.Consider
the followingexample.
Example2:
Supposethe graphof includesthe point and that the slopeof the tangentline to at any point
is givenby the expression Find
Solution: