Section 7.2: Direction Fields and Euler s Methods

Transcription

1 Sectio 7.: Directio ields ad Euler s Methods Practice HW from Stewart Tetbook ot to had i p. 5 # odd or a give differetial equatio we wat to look at was to fid its solutio. I this chapter we will eamie 3 techiques for determiig the behavior for the solutio. These techiques will ivolve lookig at the solutios graphicall umericall ad aalticall. Eamiig Solutios Graphicall Directio ields d Recall from Calculus I that for a fuctio gives the slope of the taget d lie at a particular poit o the graph of. Suppose we cosider a first order differetial equatios of the form f. or a solutio of this differetial equatio evaluated at the poit represets the slope of the taget lie to the graph of at this poit. Eve though we do ot kow the formula for the solutio havig the differetial equatio f gives a coveiet wa for calculatig the taget lie slopes at various poits. If we obtai these slopes for ma poits we ca get a good geeral idea of how the solutio is behavig.

2 Directio ields sometimes called slope fields ivolves a method for determiig the behavior of various solutios o the - plae b calculatig the taget lie slopes at various poits. Eample : Sketch the directio field for the differetial equatio result to sketch the graph of the solutio with iitial coditio.. Use the Solutio: I this problem we plot poits for the four quadrat regios ad the ad ais we will fill i the first quadrat chart i class. st Quadrat d Quadrat rd Quadrat th Quadrat

3 3 -ais ais We ca sketch the slopes o the followig graph will do i class:

4 4 Obviousl as ca be see b the last eample sketchig directio fields b had ca be a ver tedious task. However Maple ca sketch a directio field quickl. or the differetial equatio give i Eample the followig commads i Maple ca be used to sketch the directio field: > withdetools: withplots: Warig the ame chagecoords has bee redefied > de : diff^-; de : d d - > dfieldplotde color black arrows MEDIUM color blue;

5 5 Notes. The directio fields for differetial equatios of the form f where the right is strictl a fuctio of have the same slope fields for poits with the same coordiate. Slope ields are same at each coord Eample: Plot of t + cos t

6 6. The directio fields for differetial equatios of the form f where the right is strictl a fuctio of have the same slope fields for poits with the same coordiate. A differetial equatio is strictl a fuctio of the depedet variable is kow as a autoomous equatio. Slope ields are same at each coord Eample: Plot of 3. A costat solutio of the form K of a autoomous where the directio field slopes are zero that is where ad the solutio either icreases or decreases is kow as a equilibrium solutio. Eample: the equilibrium solutio is.

7 7 Eample : Give the directio field plot of the differetial equatio / 9. a. Sketch the graphs of solutios that satisf the give iitial coditios: i. iii. ii. 3 b. id all equilibrium solutios. Solutio:

8 8 idig Solutios Numericall Euler s Method A commo wa to eamie the solutio of a differetial equatios is to approimate it umericall. Oe of the more simpler methods for doig this ivolves Euler s method. Cosider the iitial value problem. over the iterval a b. Suppose we wat to fid a approimatio to the solutio give b the followig graph: Startig at the poit specified b the iitial coditio we wat to approimate to solutio at equall spaced poits beod o the ais. Let h kow as the step size be the space betwee the poits o the -ais. The + h + h 3 + h etc. Cosider the taget lie at the poit that passes through the poit. Sice the derivative is used to calculate the slope of the taget lie it ca be see that Slope of the taget lie to at

9 9 Hece lie at Slope at taget ad Slope through h h + Now cosider the lie through the poits ad. lie at Slope at taget ad Slope through h + I geeral + h Summarizig Euler s Method Give the iitial value problem we calculate from b computig h + + h where h is the step size betwee edpoits o the -ais.

10 Eample 3: Use Euler s Method with step size of.5 to estimate where is the solutio to the iitial value problem 3 4. Sketch the graph of the iterates used i fid the estimate. Solutio:

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12 Notes. Usig techiques that ca studied i a differetial equatios course it ca be show that the eact solutio to the iitial value problem 3 4 give i Eample 3 is 38 3 e The approimatio to what is whe was The eact value is e + e Thus the error betwee the approimatio ad the eact value is B decreasig the step size h the accurac of the approimatio i most cases will be better with a tradeoff i more work eeded to achieve the approimatios. or eample the chart below shows the approimatios geerated whe the step size for Eample 3 is cut i half to h.5. + h h Here the approimatio to is ad the the error betwee the approimatio ad the eact value is Eact Value 38 3 e e h e ;

13 3 The followig represets a graph of the curves produced b Euler s method for various values of h ad the eact solutio. 3. There are other umerical methods that ca achieve better accurac with less work tha Euler s method. However the uderlig approach used i ma of these methods stem from Euler s approach.