MA 366 Review - Test # 1

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1 MA 366 Review - Tes # 1 Fall 5 () Resuls from Calculus: differeniaion formulas, implici differeniaion, Chain Rule; inegraion formulas, inegraion b pars, parial fracions, oher inegraion echniques. (1) Order of a differenial equaion, normal form of a differenial equaion; general soluion; implici soluion, explici soluion; verifing a soluion; inerval of exisence. d () Direcion Fields: a soluion () o he d.e. = f(, ) has slope f(, ) ahe d poin (, ). The direcion field (or slope field) of he d.e. indicaes he slope of he angen lines o soluions a various poins (, ). The direcion field is used o give qualiaive informaion abou he behavior of soluions as (or,or, ec). Slope fields ma also be used o approximae he inerval of exisence of a soluion hrough a cerain poin. (3) SEPARABLE differenial equaions: Mehod of Soluion : 1 f() d = d dx g(x) dx = f() g(x) Warning # 1: You ma ge addiional soluions corresponding o f() =. You mus check if addiional soluions arise in his wa. Warning # : Separable equaions have soluions ha are generall given implicil raher han explicil. (4) FIRST ORDER LINEAR differenial equaions: + p() = q() Mehod of Soluion : u() () = u()q() d + C, where u() =e p() d Can now solve explicil for (). Warning # 3: The coefficien of mus be 1. (5) Mixing Problems; basic formula dx d = r ic i r c, where x() is he amoun of subsance in ank a ime ; mixing problems wih several anks (leading o a ssem of differenial equaions). 1

2 (6) Exisence and Uniqueness Theorems. These are slighl differen for nonlinear and linear differenial equaions as indicaed below. (a) NONLINEAR Differenial Equaions: Exisence Theorem: If f(, x) is defined and coninuous on a recangle R in he x-plane, hen given an poin (,x ) R, he iniial value problem x = f(, x) and x( )=x has a soluion x() defined in an inerval conaining. Uniqueness Theorem: If f(, x) and f are boh coninuous on a recangle x R in he x-plane, (,x ) R, and if boh x() and () saisf he same iniial value problem x = f(, x) and x( )=x, hen as long as (, x()) and (, ()) sa in R, we have x() =(). (The Uniqueness Theorem assers ha if f(, x) and f are coninuous on x a recangle R, hen soluions o he differenial equaion x = f(, x) canno cross in R.) (b) LINEAR Differenial Equaions: Exisence & Uniqueness Theorem for 1 s Order Linear Equaions: If p() and q() are coninuous on he inerval (α, β) which conains, hen he iniial value problem + p() = q() ( )= has a unique soluion () which exiss for α<<βand an value of. This heorem is much more precise han hose for nonlinear differenial equaions in (a). Here one can deermine he precise inerval of exisence of he soluion.

3 d (7) Auonomous Differenial Equaions: d = F () ( ) These differenial equaions are said o be auonomous since d does no depend d on he independen variable. Such equaions can have soluions ha are consan (i.e., = K) and are called equilibrium soluions. These consan soluions are found b solving F () =. Thephase line can be used o skech soluions o he d.e. ( ) and also deermine he sabili behavior of he equilibrium soluions. The sabili of equilibrium soluions o ( ) are classified as follows: (a) Asmpoicall Sable - Soluions which sar near = K will alwas approach = K as : = K (b) Asmpoicall Unsable - Soluions which sar near = K does no alwas approach = K as : = K (c) Semisable - This is a special pe of unsable soluion. In his case soluions on one side of = K will approach = K as, while soluions on he oher side of = K will approach somehing else: = K Remark: To skech non-equilibrium soluions of ( ), ou do no necessaril need direcion fields, ou can use ordinar calculus! Since d = F (), he graph d of F () vs will deermine where he soluion () isincreasing(f () > ) or decreasing (F () < ). B he Chain Rule, d df () d df () = = F (), hence d d d d agraphof df d F will deermine where he soluion () isconcaveup(f F>) or concave down (F F<). 3

4 (8) Applicaions of firs order differenial equaions: (a) Exponenial growh/deca: dn d = rn (b) Newon s Law of Cooling: dt d = k(t T e) (9) Modeling populaion growh: dp (i) Linear: d = r dp (ii) Exponenial: (iii) Logisic: d = rp dp d = rp ( 1 P K ). Pracice Problems 1. Deermine he order of each of hese differenial equaions and sae wheher he equaion is linear or nonlinear : (a) + x =1 (b) x + =1 (c) ( ) 3 + =1 (d) + =1. (a) Which of he funcions 1 () = and () = are soluions of he IVP? =, ()= (b) Which of he funcions 1 () = and () = are soluions of he IVP? =, (1)=1 3. For wha value(s) of r is = e r asoluionof 5 +6 =? 4. Find he general soluion: (a) =(e ) 3 (b) =4( +) (c) x( +1) dx d = x Find explici soluions of hese iniial value problems and deermine he inerval of exisence: (a) x =, (1) = 1 {. (b) =4( +) () = 6. (a) Find an implici soluion of he iniial value problem = x +1, (b) Find an explici soluion of he iniial value problem = ()=. x, () =. +1 4

5 7. For wha value(s) of a is he soluion of he IVP +e =, () = a bounded on he inerval? 8. Use he given direcion field o skech he soluion of he corresponding iniial value problem = f(x, ),(x )= for he indicaed iniial value (x, ): (a) (, ) (b) (, ) (c) ( 1, 3) (d) (, 4) 9. Consider he IVP : = x +, (3) = 1. (a) Is he soluion increasing or decreasing near x =3? (b) Is he soluion concave up or concave down near x =3? 1. For each of he iniial value problems deermine all iniial poins (, ) for which a unique soluion is guaraneed in some inerval conaining : (a) = +,( )= (b) =,( )= (c) = +,( )= (d) = 1/3 + 1/3,( )= (e) = 1,( )= 5

6 11. For each of he iniial value problems deermine he precise inerval of exisence: (a) = 1,(1) = (b) + (an x) =secx, () = (c) + 9 = 1,() = 1 (d) (x +4) x = 1 x,( )=1 (e) x =, (1) = 1 1. Page 61: # 1, Does he IVP dx d = (x 4)(x 1)e x, x() = 3 have a unique soluion? Consider he auonomous differenial equaion d d =8 ( 1)( 4). (a) Find all equilibrium soluions. (b) Skech soluions o he differenial equaion. (c) Classif he sabili of all equilibrium soluions. 15. Consider he auonomous differenial equaion = F (), where he graph of F () vs is shown below. (a) Find all equilibrium soluions. (b) Skech soluions o he differenial equaion. (c) Classif he sabili of all equilibrium soluions. F() Page 35: # 5, 35. Answers 1. (a) 1 s order nonlinear (b) 1 s order linear (c) 1 s order nonlinear (d) 3 rd order linear. (a) 1 and (b) 1 onl 3. r =,r=3 4. (a) = 3 e3 +C (b) = Ce4 1 Ce 4 and =and = (c)x +1 = Ce an 1 6

7 5. (a) = x, inerval of exisence x> 1 (b) =, see 4(b), inerval of x +1 exisence is << 6. (a) + = x (b) = 1+ 4x a = 1 8. See Figure below 9. (a) (3) = < so is decreasing (b) = x + +, so (3) = 3 < and so is concave down 1. (a) all (, ) (b) all (, )wih (c)all(, ) (, ) (d) all (, ) wih (e) all(, )where 1 < < 1and 11. (a) >(b) π <x<π (c) 3 <<(d) 4 <x< (e) x> 1 1. See book s answers. 13. Yes b he Exisence & Uniqueness Theorems for nonlinear equaions. Also since x =,x=, x= 1 are consan soluions, he soluion x() saisfing he d.e. wih x() = 3 is bounded: 1 <x() <. 7

8 14. (a) =, =1, = 4 (b) See figure below (c) = (semisable), = 1(sable), = 4 (unsable) 4 =4 3 1 =1 = 15. (a) = 1, = (b) See figure below (c) = 1 (unsable), = (sable) = 1 = See book s answers. 8