prelim_2_solutions

# prelim_2_solutions - Math 2930 Prelim II 7:30~9prn November...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 2930 Prelim II, 7:30~9prn, November, 2010 Answer each of the following 6 questions (20 pts. each) —-— show all work (closed book, 1 “cheat sheet”, no calculators). Academic integrity on the part of each student is presumed — Violations will be dealt with swiftly and justly. I. Consider the ODE 56 ~§~ 22E: + x 2 4cos(2r): (a) Find the general solution. (b) What is the steady—state solution (for t > 0 “large”). d2)» dy dy 2. SolvetheIVP ﬁ—-+r——4 =0, r>0, 1 :1, mm 1:0. dﬁ dz y y” dt() 3. Consider the ODE y’” + 4 y' = ﬁx). For each case, give the general form of the particular solution using the method of undetermined coefﬁcients. Do not evaluate the coefﬁcients. (a)f(x) = 5 + 8x3, (b)f(x) = xsin(5x) (c)f(x) = cos(2x), (d)f(x) = 23in2 x. 4. Consider the EV]? y" +16y = O, y(0) m y(L) = 0. (a) Find all values of L > 0 such that the BVP has a nontrivial solution (b) Determine the nontrivial solution corresponding to each of the values of L found in (a). (c) For what values of L > 0 does the BVP admit a unique solution — what is that solution? 5. Each of the following Fourier series converge to the functionsf(x) 2 x on the indicated intervals. 00 I144 co ~i . if 4 cos nx x222( srn(nx), —7r<x<7r,andx=———— Z (2 ),O\$x<n. W1 n 2 71' n=l(odd) 1’7 (a) In each case, to what value does the series converge when (1')): = "*7? f3, (ii) x = 71*, (iii) x = I In + 7: f2. ? (6 answers required in total) (b) What is the Fourier series of [x ,myr < x < 7: (period 27:)? 6. Consider the IBVP u! :uﬂ, O<x<7t, r>0, ux(0,t) 2“: ux(7r,r) = 0, u(x,0) = 5 ~+~ 3x. Obtain the steady-state solution lim n(x,r). Justify your answer. 1am 3"”‘5ﬁgﬁﬁ . “gr: W” ﬂ???“ é} MW r3?» 1%» w k3,.» M I #55:) I ﬁg 111:, - m ‘“‘ =3 : 5.2,? 5:??qu jg; _rt"" 1mm 511W MW” .e ‘p I} «:7; a, M» w.“me m.“ . was; .5“, ' -* 6‘4er ‘ WV 95: a my ‘ 3 . Wmmmmm wg-m. . WW3 Nﬁmm. _ 51% 3:: y if? win m. ...
View Full Document

## This note was uploaded on 01/21/2011 for the course MATH 2930 taught by Professor Terrell,r during the Spring '07 term at Cornell.

### Page1 / 5

prelim_2_solutions - Math 2930 Prelim II 7:30~9prn November...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online