Page 1 of 13Student Name:Student Net ID:UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGNDEPARTMENT OF MATHEMATICSMATH 286 SECTION X1 – Introduction to Differential Equations PlusFINAL EXAMINATIONDECEMBER 17, 2013INSTRUCTOR: M. BRANNANINSTRUCTIONS• This exam is three (3) hours long. No personal aids or calculators are permitted.• Answer all questions in the space provided. If you require more space to write youranswer, you may continue on the back of the page. There is a blank page at the endof the exam for rough work.• EXPLAIN YOUR WORK! Little or no points will be given for a correct answer withno explanation of how you got it. If you use a theorem to answer a question, indicatewhich theorem you are using, and explain why the hypotheses of the theorem are valid.• GOOD LUCK!PLEASE NOTE: “Proctors are unable to respond to queries about the interpretation ofexam questions. Do your best to answer exam questions as written.”Question: 1 2 3 4 5 6 7 TotalPoints: 14 25 9 9 14 15 14 100Score:Student Net ID: MATH 286 X1 Page 2 of 13SOME USEFUL FORMULAS:eB=∞Xk=01k!Bk= I + B +12!B2+13!B3+ . . .x(t) = Φ(t)Φ(a)−1x(a) + Φ(t)ZtaΦ(s)−1f(s)dsa bc d−1=1ad − bcd −b−c aan=1LZL−Lf(t) cosnπtLdt, bn=1LZL−Lf(t) sinnπtLdt1. Let f(t) be the π-periodic function defined byf(t) =(1 −π2< t < 0,t 0 ≤ t ≤π2.(a) (2 points) Sketch the graph of f over a few periods.(b) (6 points) Leta02+P∞n=1(ancosnπtL+bnsinnπtL) be the Fourier series for f. Calculatethe Fourier coefficients a0, anand bn(n ≥ 1).Student Net ID: MATH 286 X1 Page 3 of 13(c) (2 points) Does the Fourier series for f converge to f (t) at every point t? Whatdoes the Fourier series converge to when t = 0?(d) (4 points) A 1 kg cart is connected to a wall by a spring with unknown springconstant k > 0 N/m, and is periodically forced by f(t) Newtons (where f is theperiodic function defined above). Assuming there is no friction in the system, theresulting equation of motion for the displacement x(t) of the cart from rest is givenbyx00+ kx = f (t).Find all values of k that will cause resonance in the forced mechanical system.Student Net ID: MATH 286 X1 Page 4 of 132. In this multi-part problem, we will derive the solution to a one-dimensional heat equationwith mixed boundary conditions (with one endpoint held at a fixed temperature, and theother endpoint insulated). For the remainder of this problem, let L > 0 be fixed.(a) (6 points) Consider the constant functionf(x) = 100 defined on the interval [0, 2L].Sketch the graph of the 4L-periodic odd extension of f and compute its Fouriersine series.(b) (5 points) Consider the following eigenvalue problem for the function X(x) on theinterval [0, L]:X00+ λX = 0; X(0) = X0(L) = 0.Show that the λ is an eigenvalue if and only if λ = λn=(2n−1)2π24L2, where n =1, 2, 3, . . .. For each λn, write down the corresponding eigenfunction Xn(x).(Note: You may assume without proof that all the eigenvalues are positive.)Student Net ID: MATH 286 X1 Page 5 of 13(c) (3 points) A laterally insulated metal rod of length L (with thermal diffusivityk = 2) is heated to a uniform temperature of 100 degrees Celsius. At time t = 0,the left end of the rod (x = 0) is placed in an ice bath at 0 degrees Celsius, and theright end (x = L) is insulated so that no heat flows in or out at this end. If u(x, t)denotes the temperature (in degrees Celsius) of the rod at position 0 < x < L andtime t > 0, then u satisfies the one-dimensional heat equationut= 2uxx(0 < x < L, t > 0).Write down the Boundary Conditions and Initial Condition for this problem.(d) (4 points) Using the method of separation of variables, show that ifu(x, t) = X(x)T (t)is a solution to the above heat equation satisfying the boundary conditions frompart (c), then X(x) must be a solution to the eigenvalue problem in part (b).(e) (3 points) For each eigenfunction Xn(x) from part (b), find the corresponding so-lution Tn(t).Student Net ID: MATH 286 X1 Page 6 of 13(f) (4 points) Letu(x, t) =∞Xn=1αnXn(x)Tn(t).Then u(x, t) satisfies the heat equation and boundary conditions from part (c).Find the constants αnso that u(x, t) also satisfies the initial condition u(x, 0).(Hint: Use part (a).)Student Net ID: MATH 286 X1 Page 7 of 133. (9 points) Solve the initial value problemdydx=x + 3yx − y; y(1) = 0.An implicit equation for y(x) is fine.Student Net ID: MATH 286 X1 Page 8 of 134. Consider the ODEy + (2x − ey)dydx= 0.(a) (3 points) Is this equation exact?(b) (6 points) Find an implicit expression for the general solution to this ODE.(HINT: Multiply the above ODE by y and then check for exactness).Student Net ID: MATH 286 X1 Page 9 of 135. (a) (3 points) Find the general solution to the ODEy00− 10y0+ 21y = 0.(b) (6 points) Solve the initial value problemy00− 10y0+ 21y = e3x+ ex; y(0) = y0(0) = 0.(c) (5 points) Find the general solution to the ODEy(5)+ 8y(3)+ 16y0= 1 + (1 + ex) cos(2x).(NOTE: For part (c), you do not need to evaluate the undetermined coefficientsA, B, . . .).Student Net ID: MATH 286 X1 Page 10 of 136. Consider the following second order ODE for the function y(t).t2y00+ ty0+ y = 0 (t > 0).(a) (3 points) Verify that y1(t) = cos(ln t) and y2(t) = sin(ln t) are two linearly inde-pendent solutions to this ODE.(b) (3 points) Using the substitutions x1(t) = y(t) and x2(t) = y0(t), rewrite this ODEas an equivalent two-dimensional first order system of the formx0= P (t)x where x(t) =x1(t)x2(t)& P (t) =p11(t) p12(t)p21(t) p22(t).(c) (3 points) Write down a fundamental matrix Φ(t) for the system in part (b).(Hint: Use part (a).)Student Net ID: MATH 286 X1 Page 11 of 13(d) (6 points) Solve the non-homogeneous the initial value problemx0= P (t)x +t−10; x(1) =00.Student Net ID: MATH 286 X1 Page 12 of 137. The matrixA =−1 0 10 1 −40 1 −3has one eigenvalue λ with multiplicity 3.(a) (6 points) Find the eigenvalue λ and all eigenvectors associated to λ.(b) (2 points) What is the defect of this eigenvalue?(c) (6 points) Find the general solution to the 3 dimensional linear systemx0(t) =−1 0 10 1 −40 1 −3x(t).Student Net ID: MATH 286 X1 Page 13 of 13(Extra work