Lecture1Topics:

-  ReviewofDifferentialEquationsandIntegrationTechniques(7.1,7.2,and7.5)

ToDo:-  Reviewsections7.1,7.2,and7.5inthetextbook-  Tryexercisesintheseslides;workonAssignment0

DifferentialEquations

Adifferentialequation(DE)isanequationthatinvolvesanunknownfunctionandoneormoreofitsderivatives.Examples:

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y'= 2 + y y '+ 2xy = x2

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y'= x 2 + ex

DifferentialEquations

Asolutionofadifferentialequationisafunctionthat,alongwithitsderivatives,satisfiestheDE.Example:Showthatisthesolutionofthedifferentialequationwithinitialcondition.

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z(t) =1+ 1+ 2t

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dzdt

=1z −1

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z(0) = 2.

DifferentialEquations

Ingeneral,adifferentialequationhasawholefamilyofsolutions.Example:FindthegeneralsolutionoftheDE

-2.5 0 2.5 5 7.5 10

-5

5

10

y ' = 2x − 4.

DifferentialEquations

Aninitialvalueproblem(IVP)providesaninitialconditionsoyoucanfindaparticularsolution.Example:FindtheuniquesolutionoftheIVP

y ' = 2x − 4, y(0) = 3. -2.5 0 2.5 5 7.5 10

-5

5

10

Modeling:VerbalDescriptionsIVPs

Example:Writeadifferentialequationandaninitialconditiontodescribethefollowingevent:

Therelativerateofchangeofthepopulationofwildfoxesinanecosystemis0.75babyfoxesperfoxpermonth.Initially,thepopulationis68thousand..

SolutionsforGeneralDEs

Ø  AlgebraicSolutionsØ  anexplicitformulaoralgorithmforthesolution(often,impossibletofind)

Ø  GeometricSolutionsØ  asketchofthesolutionobtainedfromanalyzingtheDE

Ø  NumericSolutionsØ  anapproximationofthesolutionusingtechnologyandandsomeestimationmethod,suchasEuler’smethod

AlgebraicSolutions

Example1:Findthegeneralsolutionofthepure-timeDE

Example2:Findthegeneralsolutionofthepure-timeDE

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y'= ln x

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dydx

= 5e10x +1

1+ 25x 2

AlgebraicSolutions

Example3:FindthesolutionoftheautonomousDEwithinitialcondition

dFdt

= 0.75FF (0) = 68000.

MoreIntegrationPractice

Example:(a) (b)(c) (d)(e) (f)

x1+ x2 ∫ dx x2

1+ x2 ∫ dx

xe0.2 x∫ dx xe−x2

∫ dx

x ln x ∫ dx1x ln x

∫ dx

*Tocheckyouranswers,tryusinganonlineintegralevaluator.Forexample:https://www.integral-calculator.com/

GeometricSolutions

Example:SketchthegraphofthesolutiontotheDEgivenaninitialconditionof

y ' = arctan x

y(0) =1.

NumericSolutionsEuler’sMethodAlgorithm:AlgorithmInWords:nexttimestep=previoustimestep+stepsizenextapproximation=previousapproximation+rateofchangeofthefunctionxstepsize

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tn+1 = tn + hyn+1 = yn + F(tn ,yn )h

NumericSolutions

Example:ConsidertheIVPApproximateP(1)usingEuler’smethodandastepsizeofh=0.5.Note:WearenotabletofindanexactsolutionforthisIVP.

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dPdt

= e−t2

, P(0) = 5

NumericSolutions

Example:Calculations:

tn=tn-1+h Pn=approx.valueofsolutionattn

t0=0 P0=5

TableofApproximateValuesfortheSolutionP(t)oftheIVP

NumericSolutions

GraphofApproximateSolution:

Plotpointsandconnectwithstraightlinesegments.

tn=tn-1+h Pn=approx.valueofsolutionattn

t0=0 P0=5

t1=0.5 P1=5.5

t2=1 P2=5.9