y = (y 1)*(y 3) t

Transcription

1 MATH 66 SPR REVIEW DEFINITION OF SOLUTION A funcion = () is a soluion of he differenial equaion d=d = f(; ) on he inerval ff < < fi if (d=d)() =f(; ()) for each so ha ff<<fi: A funcion = () is a soluion of he iniial value problem d=d = f(; ); ( )= ; if = () is a soluion of he differenial equaion d=d = f(; ) on some inerval ff<<fiha conains and ( )= : DIRECTION FIELD A soluion of he differenial equaion = f(; ) has slope f( ; ) a he poin ( ; ): The direcion field of a differenial equaion = f(; ) indicaes he slope of soluions a various poins ( ; ): The direcion field gives informaion abou he behavior of soluions ha correspond o differen iniial values. The direcion field ma indicae a resricion on he domain of a soluion if he graph of he soluion appears o approach eiher averical asmpoe = or a poin ( ; ) where he graph becomes verical. Direcion fields can be ploed using MATLAB and he program dfield5. We will be using he programs dfield5, pplane5, eul, rk, and rk wih MATLAB. These were wrien b John Polking of Rice Universi. The programs are no a par of MATLAB, bu should be insalled wih MATLAB a all PUCC pc and Mac labs on campus. If ou are using a suden version of MATLAB on our own compuer, ou will need o download he files ha are appropriae for our version of MATLAB. The files can be downloaded from hp://mah.rice.edu/οdfield/. LINEAR EQUATIONS + p() = g()

2 R p() d Mulipl b he inegraing facor μ() = e, so μ = μp. Then μ + μp = μg μ + μ = μg (μ) = μg μ()() = R μ()g() d + C =( R μ()g() d + C)=μ() SEPARABLE EQUATIONS N()d=dx = M(x) R N((x))d=dx dx = R M(x) dx R N() d = R M(x) dx Equaions of he form d=dx = m(x)n() can be pu ino separable form b dividing b n(): In addiion o he soluion obained b inegraion, he differenial equaion will have consan soluions =, where is a soluion of n() =: You should know how o solve firs order differenial equaions ha are eiher separable or linear. You should be able o evaluae inegrals of he following pes: R (polmonial) dx; R e rx dx; R (ax + b)r dx, (including r = ) R (ax + b)=((x r )(x r )) dx, (parial fracions) You should be able o use given values (x )= o deermine unknown consans in a soluion. You should know he relaion of he graph of he soluion of an iniial value problem o he corresponding direcion field. MATH 66 SPR REVIEW PRACTICE QUESTIONS. Deermine he order of each of he differenial equaions; also sae wheher he equaion is linear or nonlinear. (a) + = (b) + = (c) ( ) + = (d) + p =

3 . (a) Which of he funcions () = and () = are soluions of he iniial value problem = ; ()=? (b) Which of he funcions () = and () = are soluions of he iniial value problem = ; ()=?. (a) Show ha = is a soluion of he iniial value problem = = ; () =. (b) Find a differen soluion of he iniial value problem.. For wha value(s) of r is = e r a soluion of he differenial equaion 5 +6 =? 5. Find he general soluion of he differenial equaion +(=( +) =: Assume +>. 6. Find he soluion of he iniial value problem = +; () =. 7. For wha value(s) of a is he soluion of he iniial value problem +e =; () = a bounded on he inerval? 8. Use he given direcion field of =( )( ) o deermine he behavior of as increases for each iniial value () = a: = ( )*( ) Use he given direcion field o skech he soluion of he corresponding iniial value problem for he indicaed iniial value ( ; ): Exend our skech in boh direcions as far as seems possible and explain wh he domain of he soluion ma be resriced.

4 (a) ( ; )=(; ) (b) ( ; )=(; ) = (/)*/ = ( ) ( ) Skech he direcion field of (a) = = and (b) = =:. Find an implici soluion of he iniial value problem =( )=(+ );() = :. Find an explici soluion of he iniial value problem = ; () = =: Indicae he inerval in which he soluion is valid. Find he slope of he soluion of he differenial equaion = + a he poin (; ): MATH 66 SPR REVIEW PRACTICE QUESTION ANSWERS. (a) firs order, nonlinear, (b) firs order, linear, (c) firs order, nonlinear, (d) hird order, linear. (a) and, (b). (a) = =( ) = = = ; () =, (b) =. r =; 5. = =(( + )) + =( +)+C=( +) or =( +)=+C=( +) 6. = + 7. a = 8.!as increases if a>;! as increases if a =;! as increases if a<.

5 9. (a) The graph becomes verical near (; ) =(±; ): = (/)*/ (b) The graph approaches a verical asmpoe near = : = ( ) ( ) = / = / 6 8 (a). + = +. = =( +); >. slope= 6 (b) 6 8 5