Differential Equations 2280 Midterm Exam 3 Exam Date: 22 April 2016 at 12:50pm co Instructions: This in-class exam is 50 minutes. No calculators, notes, tables or books. No answer check is expected. Details count 3/4, answers count 1/4. Chapter 3 1. (Linear Constant Equations of Order ( ii) (a) [30%] Find by variation of parameters a particular solution for the equation p” steps in variation of parameters. Check the answer by quadrature. ‘ = . Show all 2 x (b) [40%] Find the Beats solution for the forced undampeci spring-ma.ss problem x” + 256x = 231 cos(5t), x(0) = x’(O) = 0. It is known that this solution is the sum of two harmonic oscillations of different frequencies. To save time, please dont convert to phase-amplitude form. (c) [20%] Given rnx”(t) -F cx’(t) + kx(t) = 0, which represents a damped spring-mass system, assume m = 9, c = 24, k = 16. Determine, if the equation is over-damped critically damped or under-damped. To save time, do not solve for x(t). , (d) [10%] Determine the practical resonance frequency w for the RLC current equation 21” + 71’ + 501 f( z. - , = 500 sin(wt). ‘ 1 C 3I 2 Use this page to start your solution. 0 L j LJ d ia ijit - (5l II i- I’ CD cS7 ‘ n r-- 3-’ ‘1 D 4 - ‘—I _ji N - C L* —;- - ‘•= I ¶ çi I 3-, ‘ çt/ cfl C-) -- —çc- -- C) -4- I CD — I if 7 cV ci- -) C, (1 C) — €- L 4 c_p — N -+ , 3 i_ (-b’-. -i c57 cfl I if )__ 3 - 0 H 2280 Exam 3 S2016 Chapters 4 and 5 2. (Systems of Differential Equations) k(a) [30%j The 3 x 3 matrix (4 A=I 1 0 1 1 4 1 0 4 I iN has eigeiivalues A = 3, 4, 5. One Euler solution vector is fleAt with A = 3 and Find two 0) = Af(t). — = more Euler solution vectors and then display the vector general solution (t) of (b) f40%] The 3 x 3 triangular matrix (3 A=I 0 0 1 0’\ 3 1 0 4 ) represents a. linear cascade, such as found in brine tank models. Part 1. Use the linear integrating factor method to find the vector general solution (t) of t) = A(t). Part 2. The eigenanalysis method fails for this example. Cite two different methods, besides the Af(t). Don’t show linear integrating factor method, which apply to solve the system t) why the method applies. and method each precisely explain solution details for these methods, but (c) [30%] Time Cayley-Hamiltoim-Ziebur shortcut. applies especially to the system x’ =x+4y, which has complex eigcnvalues A = y’ = —4x+y, 1 ± 4i. Part 1. Show the details of the method, finally displaying formulas for x(t),y(t). Part 2. Report a fundamental matrix ((t). \O 0 o)J 75: f-’ : oo)cj r ) 0 —bcc c1—+ (0) CO C U CQ Use this page to start your solution. e: /tIe4 ( c-3=I 0 4 - -I-4 < D’ — o— —cOO .— cyc 4) 0) U -- IN 0 4) IQ) 3— — — - -U -U c-) cJ -P (34) 4) c) (_) 4) 5’ N U •1— :1- U \- ::5—) II - Th Q) - C) — Q) c.4 > Il ‘I- C) 4-) - ___—, —) ) s -) C) t C-) — (3 C-) ci) 4) ci-) (_) 4.1 rt’ - :;: 9) - - . c —— )D_ LiLt L- (Sn -, C) CJ (_) d -, ; .--j-- -) c) U -I c:i) -II jjj _7__ c) (si -- .U -U 4 C-) — — £ N U L) - -j-) 3- c) c%I c) 0 —- _\ ! ) 1 L 4 I \\ ..)//(\ ) c.)\ - 0 -4) - 1 çq L, cL.) ) 4.) c_) ti 3 C. - 0 I j/L _)_ L._i £ :,.- 0 :- 2280 Exam 3 52016 1° Chapter 6 ems) 3. (Linear and Nonlinear Dynamical Syst A 0 is stable or unstable. Then classify the ification into improper or proper 0 as a saddle, center, spiral or node. Sub-class librium [20%] Determine whether the unique equi (a) equilibrium point node is not required. = d (-1 1’\ l} d—2 (b) [30%] Consider the nonimear dynamical system xl = = An equilibrium point is x equilibrium point.. (c) 130%] —8, y 2, 2y -2V+3 2 x— — 2x(z—2y). rized system at this —4. Compute the Jacobian matrix of the linea Consider the soft nonlinear spring system { 2+ ‘ r, the phase portrait classification saddle, cente (1) Determine the stability at t = oo and bian Jaco = Ad, where A is the l system spiral or node at = for the linear dynamica matrix of this system at x = 2, y = 0. 2, y = 0 as a saddle, center, spiral or node (2) Apply the Pasting Theorem to classify x uss all details of the application of the theorem. for the nonlinear dynamical system. Disc Details count 75%. above. lusions of the Pasting Theorem used in part (c) (d) [20%] State the hypotheses and the conc Accuracy and completeness expected. f_i-a I —I— = -—2 i / -Ih’? fi js - (1 -- °LY \ ) d j (-- (I -2 \ 32. Should be -16, not 16. Jacobian in x,y is correct. Error excused. Use this page to start your solution. Un it DTh 0’ 0 — N D — + D çc ( — - a 0 I:: L> =-., — c’’=_’I c (_2 I S — — c-A C ‘-p .s. CL.. E _c L) 0 C.) “C \J’ ç— - (7 4- c) 1< c-SI 4 0 L4 •-‘ 3 C C - t. a — 4- ‘-) — 0 c 111 (cç)