Review - Quiz # 1 (1) Solving Special Tpes of Firs Order Equaions I. Separable Equaions (SE). d = f() g() Mehod of Soluion : 1 g() d = f() (The soluions ma be given implicil b he above formula. Remember, ou ma ge addiional soluions in he case of Separable Equaions. Check for division b zero.) II. Firs Order Linear Equaions (FOL). d + p() = q() Mehod of Soluion : = 1 [ µ() ] µ()q() + C, where µ() =e p() III. Homogeneous Equaions (HE). d = f(, ), where f(, ) =f(, ) OR, M(, ) +N(, ) d =, where M(, ) = n M(, ) andn(, ) = n N(, ) Mehod of Soluion : Le Subsiue hese ino d (Someimes ou ma wan o use he subsiuion = u, so ha = u and d = du + u. = f(, ) o obain he Separable Equaion du + u = F (u). = v o simplif calculaions.) IV. Eac Equaion (EE). M(, ) + N(, ) d = where M = N Mehod of Soluion : Implici soluion given b f(, ) = C where : f = M(, ) = f = f f = N(, ) == = ( M(, ) + g() ) M(, ) + g()
d () Slope Fields. A soluion = φ() o he d.e. = f(, ) has slope f(, ) ahepoin (, ). The slope field of he d.e. indicaes he slope of soluions a various poins (, ). The slope field is used o give qualiaive informaion abou he behavior of soluions as (or,or, ec). Slope fields ma also be used o approimae he inerval of definiion of a soluion hrough a cerain poin. (3) Auonomous Equaions. Equaions of he form d d = F () ( ) are said o be auonomous since d does no depend on he independen variable. Such equaions ma have soluions ha are consan (i.e., K) hese are called Equilibrium Soluions d and are found b solving F () =. You should be able o find all equilibrium soluions o differenial equaions of he form ( ) and classif he sabili of hese soluions as follows : (a) Asmpoicall Sable - Soluions which sar near = K will alwas approach = K as : = K (b) 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 no approach = K : = K
(4) Eisence and Uniqueness of Soluions. (a) THEOREM (Firs Order LINEAR). If p() andq() are coninuous on an inerval α<<βconaining, hen he IVP + p() = q() { has a unique soluion ( )= = φ() on he inerval α<<β, for an. (The larges such inerval is he inerval of definiion of he soluion.) (b) THEOREM (Firs Order NONLINEAR). If f(, ) and f are coninuous in some recangle { R: a<<b, and c<<dand (, ) lies inside he recangle R, hen he IVP = f(, ) has a unique soluion on he inerval ( )= h<< +h, for some number h>andan : d a c ( ) h +h b (To find he inerval of definiion of hese nonlinear equaions, ou mus acuall solve he IVP.) (5) Numerical Mehods. You should be able o compue b hand he firs few approimaions o he IVP d = f(, ) ( )= using he Euler (Tangen Line) Mehod : n = n 1 + hf( n 1, n 1 ) For he Euler Mehod, ou can ell if he Euler approimaion is smaller or larger han he rue soluion φ() near b looking a he sign of d a : d > a = () concave up a = EULER approimaion <φ() near d < a = () concave down a = EULER approimaion >φ() near
Pracice Problems 1. Deermine he order of each of hese differenial equaions; also sae wheher he equaion is linear or nonlinear: (a) + =1 (b) + =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. Forwhavalue(s)ofr is = e r asoluionof 5 +6 =? 4. (a) Show ha = 3 is a soluion of he iniial value problem =3 /3,()=. (b) Find a differen soluion of he iniial value problem. 5. Find an eplici soluion of he iniial value problem =, (1) = 1. Indicae he inerval in which he soluion is valid. 6. (a) Find an implici soluion of he iniial value problem = +1, ()=. (b) Find an eplici soluion of he iniial value problem =, () =. +1 7. Forwhavalue(s)ofa is he soluion of he IVP +e =, () = a bounded on he inerval? 8. Deermine wheher each of he following differenial equaions is separable, homogeneous, linear, and/or eac or none of hese. (a) + +( +3) d = (c) ( +3 +1) +( + +1)d = (e) ( +1)d +( +1) = (b) +3 +( + )d = (d) +1+( +1) d = 9. Find implici soluions o hese differenial equaions : (a) + = (b) 1+ = (c) + + = 1. Find an implici form of he general soluion of he differenial equaion d = +. 11. Find an implici soluion of he IVP +1+( +) d =, (1) = 1. 1. Consider he IVP : = +, (3) = 1. (a) Is he soluion increasing or decreasing near =3? (b) Is he soluion concave up or concave down near =3?
13. Use he given slope field o skech he soluion of he corresponding iniial value problem = f(, ),( )= for he indicaed iniial value (, ): (a) (, ) (b) (, ) (c) ( 1, 3) (d) (, 4) (e) (, 3) 14. Skech (roughl) some soluions o he auonomous equaions below. Wha are he equilibrium soluions? Classif he sabili of each equilibrium soluion: d (a) d = ( +1) ( ) (b) d d = F (), where F () isasshownbelow: w w = F() 1 1 3 4 15. For each of he iniial value problems deermine he larges inerval for which a unique soluion is guaraneed : (a) = 1, (1) = (b) + (an ) =sec, ()= (c) + 9 = 1,() = 1 (d) ( +4) = 1,( )=1
16. For each of he iniial value problems deermine all iniial poins (, ) for which a unique soluion is guaraneed in some inerval h<< + h: (a) = +, ( )= (b) = /, ( )= (c) = +, ( )= 1 (d) = 1/3 + 1/3,( )= (e) =,( )= 17. Find he eplici soluion of he iniial value problem = 1, () =. Where is his soluion defined? 18. Find he general soluion of hese differenial equaions : (a) =cos (b) e ( +) 1 d = 19. Esimae he soluion a =1.5 o he IVP = 5, (1) = using he Euler Mehod wih h =.5. Wha is he rue soluion a =1.5?. Is he Euler approimaion o he IVP d =, ( 1) = smaller or greaer han he rue soluion near = 1? 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 =3( 3 ) /3 =3 /3 ;=3() /3 (b) 5. = +1, > 1 6. (a) + = (b) = 1+ 4 +1 7. a =1 (8.) (a) HE and EE (b) HE (c) none of hese pes (d) FOL and EE (e) SE and EE 9. (a) (HE) ln 1 ( ) =ln + C and = and = (b) (SE) +1=C (c) (EE) + = C 1. 1 ( ) =ln + C 11. + + =1 1. (a) (3) = < so is decreasing (b) = + +, so (3) = 3 < andso is concave down 13. (see las page for graph) 14. (a) Equilibrium soluions = (unsable); = (sable); = 1 (semisable) (b) Equilibrium soluions = (unsable); = (semisable); = 4 (sable) (See las page for graphs) 15. (a) >(b) π <<π (c) 3 <<(d) 4 << 16. (a) all (, ) (b) all (, )wih (c)all(, ) (, ) (d) all (, )wih (e) all (, )where 1 < < 1an 17. = 1+3e 1 3e, soluion defined for 1 ln 3 <<. sin 18. (a) = + C (b) = ln ( C 1 e) 19. =.375, rue soluion φ(1.5) = 1 5 (13 58 e.5 ).396. ( 1) > hus EULER approimaion, near = 1 (Graphs on ne page...)
13. (a) 14. (b) 4 1