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# Differential Equations MTH 256

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This 6 page Class Notes was uploaded by August Feeney on Monday October 19, 2015. The Class Notes belongs to MTH 256 at Portland Community College taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/224656/mth-256-portland-community-college in Math at Portland Community College.

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Date Created: 10/19/15

MTHJSZMTHJSS Review Some De nitions amp Terminology Diuorcntial Equation DE An equation relating and unknown function and one or more ofits derivatives Ordinary Differential Equation ODE A ditferential equation that contains only ordinary derivatives 22 derivatives with respect to a single variable An ODE with independent variable and dependent variable y has the form Fvavy39vyquotv39 Where F is arealvalued function of rt 2 wr39 ab es and y39 yquot you and w are the rst second n 1 and 71m derivative of y with respect to x r r tumu r rr ii L Aquot ODE is y 25 M arch 2009 MTHJSZMTHJSS Review Some De nitions amp Terminology A 2ndorder ODE A l order ODE Flryry39ryquot 0 Gx z z 0 2 dz y m 15 0 at y independent variable MTHJSZMTHJSS Review Some De nitions amp Terminology Solution to an ODE Any function on an interval Iand possessing at least rt derivatives that are continuous on Iwhich when substituted into an VL39horder ODE results in an ident39ty 2 ODE t Zyuoy 0 lt 2 order ODE Finlay 7y 393 0 4 2 quotorder ODE A solution Egt yt 2 cos42 Another solution IE ya cos4t sin4t General solution mp yt c cos42 2 sin42 where c amp c2 are constants Kidoguchi Kenneth 25 March 2009 MTHJSZMTHJH REVRW Some De mhons amp Termmology lni alelue Framer M An ODE fur whmh mmal eehdmehs 105 are speci ed d2 ODE w f 16y n Ganeral seluueh Igt m e 5 SUMDQ eesmr Whats 5 411 are eehsms y a x zs the ODE 2 VP fl y y 2y I Famculzr seluueh Igt yo 2 cus4r ya a x zs the m MTHgszMTxLzszkmew 1 1 Differential Equauons and Mathematical Models 1 The fur mulanun uf edema mudel m mathematical arms 2 The analysis Dr seluheh quhe resulting mathemancal pmblem ungnal realrwurld sntuanun MTH szMTxLzszkmew 1 1 Mathematical Model Newton s Law of Coolmg Accurdmg m Newtun s Law quuulm The 3 h 5 Ts the budy s temperature Ts aheve n Wm mm ambient 2 Kidoquchi Kenneth MTHJSZMTHJH REVRW 1 1 Mathematical Model Example Tomcellx s Law UncleTumcelh sad that mdm an upentznk Wm uwuut ufasmall hulemthebunum wnh the velumty n Wema acquire m fenmg eely nmthewaterl e1 m Lhelevel quhehule Tumcelh s Lawcmbewntten 3 ngh where ha and m arethehexgut and vulume uf the mdm atank amme Thetank has sleek hula wnh emss secuunal area a and gxs the ty Suppuse the tank 15 anght nmular eylmaee wnh mm a 11 radms z 11 and acxrculzrleak hula ufradxus 1 mch Ifg 32 se Shaw that dh 25 March 2009 Kidoquchi Kenneth 25239MTH7253 Review MTH 1 3 Slope Fields and Solution Curves 7MTH7252 Revisited Find Fgtc if if fa where fa 41 63 a x and FH 0 F6 9 So a w 9 a Q 3 W 26 9 W y 3 9 d w 9 t Aww 25 M arch 2009 7 quot11417253 Review 1 3 Slope Fields and Solution Curves Sketch a graph of yt ify0 v and dydy 1 ry dydt 310 0 310 1 310 V2 y0 ilSZMTHJSS Review 1 3 Slope Fields and Solution Curves Considerthe ODE y39xy y01 y For a given initial condition 1c a graph ofan approximate solution can be obtaine by sketching a curve that is tangent to the slope eld maple Tool Kidoguchi Kenneth 1 3 Slope Fields and Solution Curves Existence and Uniqueness 7 Theorem 1 I 1 L 1 I in i p t t the dependent variable denoted 1 D x 6y yf y I L I L I xn y in its interior Then for some open interwl Icontaining the point x the NP dy forty Jim yo dz has on an only one solution that is de ned on the interval 1 z e a solution exists and is unique on 1 Corollary c 39 39 39 U but Dy solution toIVP exists but is not unique 25 M arch 2009 1 3 Slope Fields and Solution Curves Existence and Uniqueness 7 Theorem 1 Restated Suppose xy is a continuous function in a rectangle of the form xy l alt x lt b c lty lt in the xyplane Ifxnyu is a point in this rectangle then Val g gt 0 and a function yx de ned for xnr glt x lt xn 2 in the xyplane that is a solution to the initialValue problem dy Efampy Wu yu 1 3 Slope Fields and Solution Curves Existence and Uniqueness 7 Example y Consider the ODE if 3y xy 3yM is continuous everywhere So by the existence theorern solutions for this ODE exists A general solution is y x x c3 where c is aconstant to be determined y y0 0 d ConsidertheIVP Tray By the existence theorenn solutions for this ODE exist x L 6 6 L 3f Eb y 3 So by the uniqueness theorern solutions to the IVP need not be unique Kidoguchi Kenneth 1 3 Slope Fields and Solution Curves Existence and Uniqueness 7 Example 25 March 2009 a IVP y3ym MO 0 01 Possible solutions mm 0 y x x3 0 g o y3ltx37xgt0 0xlt1 y4quotr71zyr21 m n Kidoguchi Kenneth

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