18.03SC Final Exam = x 2 y ( ) + x This problem concerns the differential equation. dy 2

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1 803SC Final Exam Thi problem concern the differential equation dy = x y ( ) dx Let y = f (x) be the olution with f ( ) = 0 (a) Sketch the iocline for lope, 0, and, and ketch the direction field along them (c) On the ame diagram, ketch the graph of the olution f (x) What i it lope at x =? (d) Etimate f (00) (e) Suppoe that the function f (x) reache a maximum at x = a What i f (a)? (f) Ue two tep of Euler method to etimate f ( ) In (a) (c) we conider the autonomou equation x = x 3x + x 3 (a) Sketch the phae line of thi equation (b) Sketch the graph of ome olution Be ure to include at leat one olution with value in each interval above, below, and between the critical point (c) Some olution have point of inflection What are the poible value of x(a) if a non- contant olution x(t) ha a point of inflection at t = a? (d) A radioactive iotope of the element Cantabrigium, Ct, decay with half life of two year The MIT reactor run on Cantabrigium At t = 0 there i no Ct in it, but tarting at t = 0, Ct i added in uch a way that the cumulative total amount inerted by time t year i t kg Write down a differential equation for the number of mole of Ct in the reactor a a function of time What i the initial condition? dy (e) Solve the initial value problem x + 3y = x, y() = dx ( ie it ) 3 (a) Find non-negative real number A, ω, and φ uch that Re = A co(ωt φ) + i (b) Sketch the trajectory of e (i)t (c) Expre the cube root of 8i in the form a + bi (with a and b real) 4 (a) (c) Find one olution to x + x + x = q(t) for (a) q(t) = t + (b) q(t) = e t + (c) q(t) = in t What i the amplitude of the inuoidal olution? In (d) and (e), uppoe that that t 3 i a olution to x + x + x = q(t) (d) What i q(t)? (e) What i the general olution to x + x + x = q(t)? 5 (a) (b) concern the function f (t) = q(t + π ) (a) Graph f (t)

2 803SC Final Exam OCW 803SC (b) What i it Fourier erie? (Simplify the trig function) (c) Find a olution to x + x = q(t) 6 (a) (d) In a recent game of Capture the Flag, a certain tudent wa oberved to move according to the following graph, in which the hahmark are at unit pacing (a) Graph the generalized derivative v(t) (b) Write a formula for v(t) in term of the unit tep and (if neceary) the delta function (c) Still with the ame function a in (a): Graph the generalized derivative v(t) (d) Write a formula for the acceleration v(t) in term of the unit tep and (if neceary) the delta function (e) Suppoe that the unit impule repone of a certain operator p(d) i w(t) Let q(t) = 0 for t < 0 and t >, and q(t) = for 0 < t < Pleae find function a(t), b(t) o that the olution x(t) to p(d)x = q(t), with ret initial condition, i given by x(t) = b(t) a(t) w(τ) dτ 7 Thi problem concern the operator p(d) = D + 8D + 6I (a) What i the tranfer function of the operator p(d)? (b) What i the unit impule repone of thi operator? (c) What i the Laplace tranform of the olution to p(d)x = in(t) with ret initial condition? 8 In (a) and (b), A = 3 (a) What are the eigenvalue of A? (b) For each eigenvalue, find a nonzero eigenvector (c) Suppoe that the matrix B ha eigenvalue and, with eigenvector and repectively Calculate e Bt

3 803SC Final Exam OCW 803SC (d) What i the olution to u = Bu with u(0) =? 9 (a) Suppoe again that the matrix B ha eigenvalue and, with eigenvector and repectively Sketch the phae portrait on the graph below a (b) Let A =, and conider the homogeneou linear ytem u = Au For each of the following condition, determine all value of a (if any) which are uch that the ytem atifie the condition (i) Saddle (ii) Star (iii) Stable node (iv) Stable piral What i the direction of rotation? (v) Untable piral (vi) Untable defective node { x = x y 0 Part (a) (c) deal with the nonlinear autonomou ytem y = x + y 8 (a) Find the equilibria of thi ytem (b) There i one equilibrium in the outh-wet quadrant Find the Jacobian at thi equilibrium (c) The equilibrium you found in (b) i a table piral For large t, the olution which converge to thi equilibrium have x-coordinate which are well-approximated by the function Ae at co(ωt φ) for ome contant A, φ, a, and ω Some of thee contant depend upon the particular olution, and ome are common to all olution of thi type Find the value of the one which are common to all uch olution (d) Finally, return to the autonomou equation x = x 3x + x 3 that you tudied in problem Write down a formula approximating the olution converging to the table equilibrium when t i large Operator Formula Exponential Repone Formula: x p = Ae rt /p(r) olve p(d)x = Ae rt provided p(r) = 0 Reonant Repone Formula: If p(r) = 0 then x p = Ate rt /p (r) olve p(d)x = Ae rt provided p (r) = 0 Defective matrix formula If A i a defective matrix with eigenvalue λ and nonzero eigenvector v, then you can olve for w in (A λ I)w = v and u = e λt (tv + w) i a olution to u = Au Propertie of the Laplace tranform 3

4 803SC Final Exam OCW 803SC 0 Definition: L[ f (t)] = F() = f (t)e t dt for Re >> 0 0 Linearity: L[a f (t) + bg(t)] = af() + bg() Invere tranform: F() eentially determine f (t) 3 -hift rule: L[e at f (t)] = F( a) { f (t a) if t > a 4 t-hift rule: L[ f a (t)] = e a F(), f a (t) = 0 if t < a 5 -derivative rule: L[t f (t)] = F () 6 t-derivative rule: L[ f (t)] = F() [generalized derivative] L[ f (t)] = F() f (0+) [ f (t) continuou for t > 0] r t 7 Convolution rule: L[ f (t) g(t)] = F()G(), f (t) g(t) = f (τ)g(t τ)d τ 0 8 Weight function: L[w(t)] = W() =, w(t) the unit impule repone p() Formula for the Laplace tranform n! L[] = L[e at ] = L[t n ] = a n+ ω L[co(ωt)] = L[in(ωt)] = + ω + ω e a L[u a (t)] = L[δ a (t)] = e a where u(t) i the unit tep function u(t) = for t > 0, u(t) = 0 for t < 0 Fourier erie a 0 f (t) = + a co(t) + a co(t) + + b in(t) + b in(t) + π a m = f (t) co(mt) dt, b m = f (t) in(mt) dt π π co(mt) co(nt) dt = in(mt) in(nt) dt = 0 for m = n co (mt) dt = in (mt) dt = π for m > 0 If q(t) i the odd function of period π which ha value between 0 and π, then ( ) 4 in(3t) in(5t) q(t) = in(t) π 3 5 4

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