DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations

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1 DIFFERENTIAL EQUATIONS wih TI-89 ABDUL HASSEN and JAY SCHIFFMAN W will assum ha h radr is familiar wih h calculaor s kyboard and h basic opraions. In paricular w hav assumd ha h radr knows h funcions of h SECOND, APLHA, and GREEN DIAMOND kys. Thus w will simply say display h Y=dior assuming ha h radr will firs prss h DIAMOND ky and hn F. W hav us bold fac uppr cas lrs o rfr o h calculaor s commands or kys. Whn w say You can accss h command and from Mah 8 8, w man firs prss MATH. (which is nd hn 5) and hn prss 8 wic. No ha mos mnus hav svral submnus, which in urn may hav many opions. To display and slc an opion w may nd o us h cursr kys. Mos buil-in funcions can b found by prssing CATALOG followd by h firs lr of h dsird command. On can hn slc h command by scrolling down, if ncssary, using h cursor ky. To clar h hom scrn, us F 8. To go back o h hom scrn, us HOME or ESC or QUIT. To obain an approxima valu (in dcimals) us h GREEN DIAMOND followd by ENTER. To draw a graph us GREEN DIAMOND followd by F3. A. Dircion Filds and Graphs of Diffrnial Equaions Exampl : Draw h dircion fild of h diffrnial quaion y ' = x! y Soluion:. Prss h MODE ky and from h GRAPH mod slc 6: DIFF EQUATIONS.. Display h Y= dior and nr your diffrnial quaion. Us for indpndn variabl and y for y. For h prim noaion for h drivaiv us ND =. Draw h graph. Th figurs blow show h abov sps. Exampl : Draw h graph of h soluion of y ' x y =! ha passs hrough (,-) Soluion: Enr your diffrnial quaion as in Exampl.. Us h cursor ky o mov up o h iniial valu of and prss F3. Thn yp and ENTER.. Typ for h y-valu in h lin y i =. Draw h graph. Th firs figur blow shows h inpus whil h scond shows h graph. Rmarks:. To draw h graph of h soluion wihou h dircion fild, from h Y= dior prss F 9 o display h GRAPH FORMAT. From h Filds opion (h las row) us h righ cursor ky o choos 3:FLDOFF and prss ENTER. Now draw h graph.

2 . You can obain soluions passing hrough ohr poins by simply changing h iniial condiions whil you ar in h dircion fild/graph scrn. To do his prss F8 (which is ND F3.) and yp h iniial condiion for and ENTER and hn yp h iniial valu for y and ENTER. 3. Thr ar svral syls for h graph of h soluion hrough a givn poin. From h Y= dior scrn prss F6 (which is nd F) and choos any of h syls and s wha happns. Th figurs blow show GRAPH FORMAT and h graph of h quaion y ' = x! y wih diffrn iniial valus using F8 (from h graph display). Exampl 3: Draw h dircion fild for h sysm of diffrnial quaions! dx = 5x + 3y " d dy = 4x " 3y " d Soluion: In h Y= dior us y for x and y for y. From h Filds opion of h graph forma, choos :DIRFLD. (S Rmark abov.) Th firs wo figurs show h inpus and h dircion fild, rspcivly. Th las wo figurs show a graph of an iniial valu problm for sysms of quaion. (Wri h IVP.) Exampl 4: Draw h dircion fild for y '' + 3 y '! y = x. Soluion. TI-89 draws dircion filds only for firs ordr and sysms of firs ordr diffrnial quaions. Thus w nd o convr his scond ordr quaion in o sysms of firs ordr quaions. As bfor w us for h indpndn variabl and y for y. W l y ' = y Thn h givn quaion is quivaln o h sysm! dy = y " d " dy = + y 3y " d Procd as in Exampl 3. Th main poin of his xampl is ha w can us his chniqu for highr ordr diffrnial quaions. (S Scion D blow) B. Solving Firs and Scond Ordr Diffrnial Equaions Th command for solving firs and scond diffrnial quaions is dsolv( which can b accssd by F3 C. Th forma for his command is dsolv(h diffrnial qn, indpndn variabl, dpndn variabl) Exampl 5: Solv a) Soluion: = + b) y '' + 5 y '! 6y = 0 y ' y x In h inpu lin nr dsolv( y ' = y + x, x, y ) ENTER. Th figurs ll h sory!

3 No ha TI 89 is giving you h gnral soluion for a) as y x " x " x ". sands for an arbirary paramr (ha w usually wri as c or c.) x Exampl 6: Solv h iniial valu problm (IVP) y '' + y '! 3y =, y (0) = and y '(0) =! x Soluion: In h inpu lin nr dsolv( y '' + y '! 3y = and y (0) = and y '(0) =!, x, y) ENTER You can accss and from Mah ( which is nd 5) 8 8. You could also yp i. Us ALPHA (-) for spac. x Exampl 7: Draw h graph of h soluion of h IVP y '' + y ' + 5y =, y (0) = and y '(0) =! Soluion: Firs solv h IVP as in Exampl 6.Thn from h inpu lin us F4 o Dfin y(x) as h soluion. You can g h soluion by using h upward cursor ky and prssing ENTER. Mak sur o dl y. This will auomaically nr h soluion as y on h Y=dior. Hr is h soluion and h graph. (W hav usd [-5,8] by [-0,30] for h graph window.) Mak sur ha h MODE is on Funcions no on Diff Equaions. C. Eulr and Rung-Kua Mhods Th TI 89 can gnra numrical soluions using h Eulr and Rung-Kua mhods. Th command for his is BldDaa nam. Hr BldDaa is h command for building h abl of valus and nam is h nam of h daa. W show h dails in h following xampl. Exampl 8: Considr h IVP: y ' = x! y, y (0) = a) Us Eulr s mhod o consruc a numrical soluion for h IVP. b) Us Rung-Kua mhod o consruc a numrical soluion for h IVP c) Solv h IVP d) Consruc a abl o compar h wo numrical mhods and h xac soluion. Soluions: a) Hr ar h sps o build h daa for h numrical soluion using Eulr s mhod.. Enr h diffrnial quaion in h Y= dior and dl any ohr quaions.. From h graph forma, slc EULER for Soluions Mhod and FLDOFF for Filds. 3. Prss HOME hn CATALOG and b. Slc BldDaa and ENTER. Typ u for h nam and ENTER. (You could also yp blddaa u on h inpu lin afr your prss HOME) 4. Opn u using APPS 6 and hn slc u for Variabl. b) For h Rung-Kua mhod from h graph forma, slc RK for Soluions Mhod. Prss HOME and yp blddaa rk and ENTER.

4 c) Solv h IVP using dsolv( and Dfin y(x) as h soluion.(s Exampl 7.) d) To compar h wo mhods and h xac soluion, follow hs sps.. Us APPS 6 3 o cra a nw daa, call i comp.. Prss F4 and yp u[] and ENTER.(u[] rfrs o h firs column of h daa calld u.) 3. Mov o h scond column and prss F4. Typ u[] and ENTER. 4. Mov o h hird column and prss F4. Typ rk[] and ENTER. 5. Mov o h fourh column and prss F4. Typ y(c) and ENTER. Hr ar h figurs showing h rsuls of h abov sps. If your abl is diffrn from ours, chang h valu of sp in h WINDOWS o 0.. D. Solving Third and Highr Ordr Diffrnial Equaions Rmark: TI 89 dos no solv 3 rd and highr ordr diffrnial quaions. To obain h graph of a soluion of hird and highr ordr quaion, w convr h quaion ino sysms of firs ordr quaions and draw h graphs.(s Exampl 4 abov.) Howvr, w can uiliz h TI 89 capabiliy o solv polynomial quaions wih complx roos o solv linar diffrnial quaions of highr ordr wih consan cofficins. Hr ar som xampls. Exampl 9: Solv y ''' + 3 y ''! y '! 3y = 0 Soluion: Th auxiliary quaion is 3 + 3!! 3 = 0 and using h csolv( (which can b accssd by F A ) command for solving quaions wih complx roos, w obain = or =! or =! 3. Thus h x! x! 3x gnral soluion is givn by y = c + c + c3. Exampl 0: Solv y ''' + 3 y '' + 8 y ' + 6y = 0. Soluion: Th auxiliary quaion is = 0 and using csolv( = 0, ) w g! x! x! x =! + 5i or =!! 5i or =!. Thus h gnral soluion is y = c + c cos(5 x) + c3 sin(5 x) Exampl : Solv h IVP y '''! y ''! 4 y ' + 4y = 0, y (0) =! 4, y '(0) =!, and y ''(0) =! 9 Soluion: Th auxiliary quaion hr is 3!! = 0 and csolv( 3!! = 0, ) yilds h soluions = or = or =!. Now us F4 o dfin h gnral soluion as y = a " x + b " x + c "! x. To solv for a, b, c using h iniial condiion, w could ENTER Solv( ( y x = 0) =! 4 and ( d( y, x) x = 0) =! and ( d( y, x,) x = 0) =! 9,a ) For h drivaiv, us F3 or nd 8. Hr ar h sps.

5 Thus h soluion of h IVP is y =! 3 x + x!! x. E. Solving Sysms of Diffrnial Equaions In Scion A w hav discussd how o obain h graph of a soluion of a sysm of diffrnial quaions. Hr w will solv sysms wih consan cofficins using h hory of ignvalus and ignvcors. Exampl : Solv h sysm of quaions givn by X ' = AX whr "! 3 A =! Soluion: I is now rcommndd ha you clar all h singl variabls you migh hav usd arlir. F6 ENTER will accomplish his. Hr ar h rlvan sps.. Th firs ask will b o nr h marix A. Us APPS 6 3 and Typ choos :marix. Us h down cursor ky o go o h variabl box and yp a for h nam of h marix. For boh row and col dimnsions yp.(again us h down cursor ky afr you ypd h inpus.)now ENTER and yp h nris of h marix.(th firs row mus b filld in firs.) Prss HOME and CLEAR. Find h ignvalus of h marix by using Mah 4 9 a) ENTER or by yping igvl(a) 3. Find h ignvcors of h marix by using Mah 4 A a) ENTER or by yping igvc(a) Th figurs blow ar h rsul of h abov sps. Th las figur shows h ignvalus and vcors. Thus h gnral soluion of h quaion is givn by " " X c c! =! !.368 Rmark:. Th firs numbr givn by igvl(a) is h firs ignvalu which in his cas is and scond ignvalu is. Th firs column of h igvc(a) is an ignvcor corrsponding o h firs ignvalu of a. No ha TI 89 is normalizing h vcors, ha is h ignvcors ar uni vcors.. For our purposs and asir noaions, i is convnin o rwri h ignvcors wih ingr nris. This is usually possibl. On possibl mhod is o rplac h smalls numbr in h columns by and divid h ohr nris in ha column by h smalls valu you jus rplacd. Us h command igvc(a)[j,k] o rfr o h j-k nry of h marix igvc(a).i is clar ha h firs columns ar qual! " hus for h fis ignvcor w may ak. Th scond on may no b clar so w rplac & by. No hn ha /-.368 is Thus i is highly rcommndd ha you compu 3 igvc(a)[,]/ igvc(a)[,]. W find ha his is 3. Thus w may ak! " as h scond ignvcor. & Thus h gnral soluion could also b givn by "! " 3. X = c + c

6 ! 3. No ha w can xprss h abov soluion as 3 c X " " = c! & ' somims rfrrd o as h fundamnal marix of h quaion!. Th marix 3 X ' = AX. " b =! & ' is Exampl 3: Solv X ' = AX,! " X (0) =, whr! 3 " A =! Soluion: As in Exampl w solv X ' = AX and xprss h answr in h form givn in Rmark 3! " abov. All w hav o do now is solv h sysm of quaions 3 " c "!! c = 0 =! To his nd w nr h marix " 3 ino h calculaor as b. and "! as d. Th w compu h! & ' command rrf(augmn((b =0),d)). Th las column of his row rducd chlon form marix givs h soluion for c and c. rrf and augmn can b accssd from Mah( nd 5) 4 4 and Mah 4 7, rspcivly. Exampl 4: Solv X ' = AX + F, whr! " F = &! c " c = c & Soluion: L b b as in Exampl and l and! 3 " A =!. Thn h gnral soluion o h sysm is givn by!! X = b" c + b" ( b " f ) d. Enr F as f and xcu h command b " ( b " f ) d. For!, us F3 or nd 7. Hr ar h figurs for hs sps. Thrfor h gnral soluion is givn by Exampl 5: Solv X ' AX F " 3! + 3! " 3 c " ( 4 ) * + X = (! ) ( c ) + ( ), -, - ( ) (&! ' +! ),* 4 + -! " X (0) =, whr! " and "! 3 F = A = &! = +, Soluion: W will us h noaions of Exampls 3 and 4. Th soluion is hn givn by h formula:!! ( (0)) ( ( ) ( )) X = b" b " d + b" b s " f s ds W now nd o Dfin b^(-) and f as funcions of s rahr han as funcions of. W will us and g, for b^(- ) and f, rspcivly. Hr is a parial picur. 0. Th soluion is!! " X =! + + 3,! + +! & '