Mah Spring 6 Differenial Equaions Final Exam Due Dae: Tuesday, June 6, 5pm Your name (please prin): Insrucions: This is an open book, open noes exam. You are free o use a calculaor or compuer o check your answers, bu you mus jusify all of your answers o receive credi. You have unil he due dae o work on all 9 problems. However, he exam is no inended o ake more han o 4 hours o complee. The Honor Principle requires ha you neiher give nor receive any aid on his exam. Addiionally, he only resources you may consul while aking he exam are he course ex book and your personal noes, exams and homework. Problem 4 5 6 7 8 9 Toal Poins 5 5 5 5 5 5 5 Score
Problem. Consider he non-linear auonomous sysem x = x x = x x + x x. a. Find all of he criical poins of his sysem and classify hem as o sabiliy and ype. b. Show ha he non-linear second order differenial equaion y + (y ) + y + y y = can be ransformed ino he firs order sysem above. c. Use pars (a) and (b) o describe he long erm behavior of soluions o he iniial value problem y + (y ) + y + y y = y( ) = y y ( ) = y when y and y are sufficienly small. Problem. a. Find an auonomous ordinary differenial equaion wih an unsable criical poin a y = and an asympoically sable criical poin a y =. b. Find an auonomous ordinary differenial equaion wih a semi-sable criical poin a y = and an unsable criical poin a y =. Problem. Find he general soluion o he sysem x = 4 x. Classify he criical poin a he origin as o sabiliy and ype, and skech he phase porrai.
Problem 4. Le he funcion f be defined on he inerval [, L] by L x < L/4, f(x) = L/ L/4 x < L/, L/ x L. a. Find he Fourier cosine series for f wih period L. Skech he graph of he funcion o which he Fourier series converges. b. Use par (a) o solve he boundary value problem u xx = 9u, < x <, > u x (, ) = u x (, ) =, > x < 5, u(x, ) = 5 5 x < 5 5 x. [Hin: This boundary value problem corresponds o a cerain ype of hea conducion problem. Which one is i?] Problem 5. Consider he nonlinear differenial equaion a. Show ha he subsiuion dy d + = y y. () ransforms () ino he linear equaion y = + u du d + u =. () b. Solve equaion () and use his o find he general soluion o ().
Problem 6. This problem deals wih he nonlinear auonomous sysem dx d dy d = y = x. a. Find an equaion saisfied by he rajecories of his sysem. b. Skech several of he rajecories found in par (a). Be sure o indicae he direcion of moion along hese rajecories. c. Use par (b) o deermine he ype and sabiliy of his sysem s criical poin a he origin. Problem 7. Find all soluions of he boundary value problem u xx + u yy =, < x < L, < y < u(, y) = u(l, y) =, < y < which are of he form u(x, y) = X(x)Y (y). Problem 8. Mach he following differenial equaions wih heir direcion fields (shown on he nex page). You do no need o show any work. a. y = + b. y = y c. y = y d. y = y + e. y = sin( y) f. y = ( + y )
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Problem 9. Consider he second order equaion xy + y + xy =, x >. () a. Show ha x = is a regular singular poin of (). Furhermore, show ha () has soluions of he form y = x r a n x n only if r = or r =. b. Show ha when r = boh a and a are free, and find he recursion relaion saisfied by he remaining coefficiens. c. Show ha when we se a = and a = in (b) we ge he soluion n= y (x) = cosx x and when we se a = and a = we ge he soluion y (x) = sin x x.