Math 251 – Sections 1/2December 14, 2005 Final ExamNamePlease check one of the boxes below Section 1 – 1st per – 8:00am Section 2 – 3rd per – 10:10amThere are 10 questions on this exam. Question 2, which has 12 parts, is worth 24 points. The otherquestions are worth 14 points each. The total number of points is 150. If a question has multipleparts, then the points assigned to the question are divided equally among the parts, unless otherwiseindicated.Where appropriate, show your work to receive credit; partial credit may be given.Please turn your cell phone OFF.The use of calculators, books, or notes is not permitted on this exam.Time limit 1 hour and 50 minutes.1. Consider the functionf(x) =0 if x < 03 if 0 ≤ x < 10 if 1 ≤ xWrite down the first seven terms in the Fourier series of f(x) on [−2, 2].2. In Parts a. through e. sn(x) denotes the nth partial sum of the Fourier series in Problem 1.a. Find limn→∞sn(4)b. Find limn→∞sn(5)c. Find limn→∞sn(6)d. Is it true that for all sufficiently large n: sn(x) ≥ −0.15 for every x in [1.4, 1.7] ?e. Is it true that for all sufficiently large n: sn(x) ≤ 3.15 for every x in [0.9, 1.1] ?f. What are the values of the following integrals?Z6−6cosπ3xcos5π6xdxZ6−6sinπ6xsinπ6xdxg. What are the values of the following integrals?Z60cosπ6xdxZ6−6sinπ17xdxh. If u(x, t) is the temperature of a thin rod of length L insulated on its sides and ends, thenwhat are ux(0, t) and ux(L, t) for any t?i. We can find a sine series for the function f(x) = x on the interval [0, 2]. To what value doesthe sine series converge at x = 2?j. We can also find a cosine series for the function f(x) = x on the interval [0, 2]. To whatvalue does the cosine series converge at x = 2?k. One of the following two questions is much easier than the other. Answer the easier one.I. What is the sine series for f(x) = 7 on [0, 4]?II. What is the cosine series for f(x) = 7 on [0, 4]?l. Supp ose that a car does not pass the safety inspection because it crosses its equilibriumposition more than once after being shaken up and down. Also suppose that there is a severeshortage of sho ck absorbers but an abundance of springs in the auto parts stores. So installingnew shocks is not an option but would exchanging the current set of springs with ones thathave a larger spring constant enable the car to pass the inspection?3. Consider the two point boundary value problemy00+ λy = 0 y0(0) = 0, y0(2) = 0You may assume that there are no eigenvalues with λ < 0.a. 2pt Write down the general solution of the above ODE when λ = 0b. 2pt Using your answer to Part a. determine whether or not λ = 0 is an eigenvalue. If itis, then what is the corresponding eigenfunction?c. 2pt Write down the general solution of the above ODE when λ > 0d. 6pt Using your answer to Part c. determine all eigenvalues and eigenfunctions withλ > 0.e. 2pt Which one of the following partial differential equations can be solved by using thetechnique of separation of variables? Circle it:ut= ux+ 1 ut= ux+ u4. Supp ose a thin homogeneous rod 5 cm long is insulated along its sides and made of a materialwith thermal diffusivity α2= 0.8 and that the left end is held at 10oand the right end is heldat 60o,a. 4pt Find the initial temperature distribution that leads to a steady state solution to theabove problem (ie, a solution that is constant with respect to t.b. 6pt Consider the same problems as in Part a. EXCEPT that the initial temperaturedistribution isf(x) = 10Find the temperature u(x, t) of the rod at any time t > 0 and at any point x inside the rod0 < x < 5. (If the answer involves finding a sine or cosine series then DO NOT find theactual values of the anand/or bnwhich appear in the answer.)c. 4pt Consider the same problem as in Part b.. After a long time passes, approximatelywhat is the temp erature of the rod at x = 3 cm?5. a. 4pt Suppose a thin homogeneous rod 5 cm long is insulated along its sides and madeof a material with thermal diffusivity α2= 0.8. Assume that also the ends of this rod areinsulated. and that the initial temperature distribution of the rod isg(x) = x3Find the limit of the temperature u(x, t) as t → ∞.b. 4pt The displacement u(x, t) of a string of length 5cm with ends clamped satisfies thedifferential equation 4uxx= utt. If the initial displacement of the string is 0 and the initialvelocity is given by sin(2πx), then what is the displacement u(x, t) of the string at t > 0.c. 6pt Now assume that the initial displacement of the string in Part b is sin(5πx) andthe initial velocity of the string is 0. Write down a formula for the displacement of the thedisplacement u(x, t) of the string at t > 0.6. a. 4pt Find the solution of the Laplace equation on the rectangle{(x, y) | 0 < x < 5, 0 < y < 7}which has the following values on the boundary:u(0, y) = 0 if 0 < y < 7u(x, 0) = 0 if 0 < x < 5u(x, 7) = 0 if 0 < x < 5u(5, y) = 4 sin(3πy) if 0 < y < 7b. 4pt Consider the function F (θ) = 11 + 10 cos(9θ) + 8 sin(7θ) defined on [−π, π] Find thesolution to the Dirichlet problem for the unit disk with the values on the boundary given byF (θ). That is, find a function in polar coordinates u(r, θ), which is harmonic when r < 1 andwhich satisfies the following when r = 1:u(1, θ) = F (θ ) for − π < θ < πc. 4pt If the function F (θ) in Part b. is replaced by F (θ) = |θ| then find the find the valueof the solution to the Dirichlet problem at the center of the disk: u(0, 0).d. 2pt Now suppose that only thing known about the p.w. continuous function F (θ) in Part b.is that its values are between between 10oand 20o. If u(0, 0) = 20, then what is u(1/2, π/2)?7. a. Solve the following IVP. (DO NOT use Laplace transforms.)y00− 4y0+ 4y = 0 y(1.23) = 1, y0(1.23) = −1b. Find a particular solution to the following nonhomogeneous linear ODE (using any methodcovered in Math 251):y00− 4y0+ 4y = e2t8. In each part of this Problem do the following:i. Sketch a phase portrait for this system.ii. State the name associated with the critical point at (0, 0) and state whether it is stable,asymptotically stable or unstable?a. 5pt The homogeneous linear system x0= Ax whose general solution is:x = c1e−4t11+ c2e−2t1−1b. 5pt The homogeneous linear system x0= Ax, where A =2 30 2.c. 4pt The homogeneous linear system x0= Ax, where A =2 00 −2.9. Solve the following initial value problems:a. ty0= 2t2e2t+ y
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