CK12 Calculus - Single Variable

(Marvins-Underground-K-12) #1
Let be sequencedefinedsuccessivelyby,

for
The sequenceof approximationsconvergesto the solution , i.e.

Now that we havedefinedPicard’s method,let’s calculatea sequenceof functionsfor an initialvalueproblem.
Example 1


Find the first four functions definedby Picard’s methodfor the solutionto the initialvalue
problem
y(x) = xy(x) with y(-1) = 1.
Solution
We want to applythe FundamentalTheoremof Calculusto the differentialequationsso that it is reformulated
for use in the Picardmethod.Thus,


Now that the differentialequationhas beenrewrittenfor Picard’s method,we beginthe calculationsfor the
sequenceof functions.In all casesthe first functionY 0 (x) is givenby the initialcondition:


Step 1 – DefineY 0 (x) = 1


Step 2 – SubstituteY 0 (x) = 1 fory(t) in the integrandof :

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