# Ch 1.2: Solutions of Some Differential Equations

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1 Ch 1.2: Solutions of Som Diffrntil Equtions Rcll th fr fll nd owl/mic diffrntil qutions: v 9.8.2v, p.5 p 45 Ths qutions hv th gnrl form y' = y - b W cn us mthods of clculus to solv diffrntil qutions of this form. Clssify first ordr DE: (1) Intgrbl qutions (2) Sprbl qutions (3) Linr qutions

2 Intgrbl Equtions (Exmpls) y 2 ' t 1 y' sin( t) (1) (2) 2t y' 5 y' y2 (3) (4) Qustion: Cn w us th sm pproch for th fourth qution? It is clld sprbl qution.

3 Sprbl Equtions Qustion: Is thr ny wy to trnsform th 4 th qution into som qution clos to intgrbl qution? y' y2 () n quilibrium solution: y(t) = 2 (b) W ssum tht y is not n quilibrium solution. (Exmpls) Initil Vlu Problm (IVP) y' 2y 5, y() y (1) Find n quilibrium solution (2) Find gnrl solution nd th solution of th IVP.

4 Exmpl 1: Mic nd Owls (1 of 3) To solv th diffrntil qution dp dt w us mthods of clculus, s follows. dp dt ln.5 Thus th solution is p 9 p 9 p 9.5 p 45 p 9 c whr c is constnt..5t C.5t.5t C dp / dt p 9.5 p 9 p 9 c.5t dp p 9.5tC, c C.5dt

5 Exmpl 1: Intgrl Curvs (2 of 3) Thus w hv infinitly mny solutions to our qution, p.5 p 45 p 9 sinc k is n rbitrry constnt..5t c Grphs of solutions (intgrl curvs) for svrl vlus of c, nd dirction fild for diffrntil qution, r givn blow. Choosing c =, w obtin th quilibrium solution, whil for c, th solutions divrg from quilibrium solution.,

6 Exmpl 1: Initil Conditions (3 of 3) A diffrntil qution oftn hs infinitly mny solutions. If point on th solution curv is known, such s n initil condition, thn this dtrmins uniqu solution. In th mic/owl diffrntil qution, suppos w know tht th mic popultion strts out t 85. Thn p() = 85, nd p( t) 9 c.5t p() 85 5 Solution : 9 c c p( t) 9 5.5t

7 Solution to Gnrl Eqution To solv th gnrl qution (: sprbl qution) y y b w us mthods of clculus, s follows. dy dt y ln b y b / y b / t C Thus th gnrl solution is t b t y c, whr c is constnt. C dy / dt y b / y y b / b / c t dy y b /, tc c C dt

8 Initil Vlu Problm Nxt, w solv th initil vlu problm y y b, y() y From prvious slid, th solution to diffrntil qution is y b t c Using th initil condition to solv for c, w obtin b b y() y c c y nd hnc th solution to th initil vlu problm is y b y b t

9 Equilibrium Solution To find th quilibrium solution, st y' = & solv for y: st b y y b y( t) From th prvious slid, our solution to th initil vlu problm is: Not th following solution bhvior: If y = b/, thn y is constnt, with y(t) = b/ y b y If y > b/ nd >, thn y incrss xponntilly without bound If y > b/ nd <, thn y dcys xponntilly to b/ If y < b/ nd >, thn y dcrss xponntilly without bound If y < b/ nd <, thn y incrss symptoticlly to b/ b t

10 Exmpl 2: Fr Fll Eqution (1 of 3) Rcll qution modling fr fll dscnt of 1 kg objct, ssuming n ir rsistnc cofficint = 2 kg/sc: dv / dt v Suppos objct is droppd from 3 m. bov ground. () Find vlocity t ny tim t. (b) How long until it hits ground nd how fst will it b moving thn? For prt (), w nd to solv th initil vlu problm v 9.8.5v, v() Using rsult from prvious slid, w hv y b y b t t.2t v v

11 Exmpl 2: Grphs for Prt () (2 of 3) Th grph of th solution found in prt (), long with th dirction fild for th diffrntil qution, is givn blow. v 9.8.5v, v 49 1 t /5 v()

12 Exmpl 2 Prt (b): Tim nd Spd of Impct (b) Nxt, givn tht th objct is droppd from 3 m. bov ground, how long will it tk to hit ground, nd how fst will it b moving t impct?

13 Exmpl 2 Prt (b): Tim nd Spd of Impct (3 of 3) (b) Nxt, givn tht th objct is droppd from 3 m. bov ground, how long will it tk to hit ground, nd how fst will it b moving t impct? Solution: Lt y(t) = distnc objct hs flln t tim t. It follows from our solution v(t) tht Lt T b th tim of impct. Thn y'( ) ( ) ( ) t/5 t/5 t v t y t t C t/5 y() C 245 y( t) 49t s( T) 49T 245 T / Using solvr (root-finding sw), T 1.51 sc, hnc v(1.51) (1.51) 43.1ft/sc

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