Math 251December 14, 2005 Final ExamNameSectionThere are 10 questions on this exam. Many of them have multiple parts. The point value of eachquestion is indicated either at the beginning of each question or at the beginning of each part wherethere are multiple partWhere appropriate, show your work to receive credit; partial credit may be given.The use of calculators, books, or notes is not permitted on this exam.Please turn off your cell phone.Time limit 1 hour and 50 minutes.Question Score1 18pt2 16pt3 12pt4 14pt5 12pt6 14pt7 14pt8 16pt9 20pt10 14ptTotal 150pt1. Consider the nonlinear system:x0= 2x − xyy0= −3y + xya. 2pt Find all the critical p oints of the nonlinear system.b. 6pt In a neighborhood of each critical point approximate the nonlinear system by a linearsystem.c. 2pt Determine the name and the stability of the the critical points of each of the linearapproximations.d. 2pt Sketch a phase portrait for the original nonlinear system.e. 2pt The linearization of a nonlinear system may have a critical point that is not guaranteedto reflect the behavior of the original nonlinear system. List the three types of critical pointsfor which this may happen.2. Consider the functionf(x) =0 if x < 03 if 0 ≤ x < 10 if 1 ≤ xa. 10pt Find the Fourier series of f (x) on [−2, 2]. (Either use summation notation to writethe answer or write the first seven terms.)In Parts b. through d. sn(x) denotes the partial sums of the Fourier series on [−2, 2] of thefunction f(x).b. 2pt Find limn→∞sn(7)c. 2pt Find limn→∞sn(8)d. 2pt Find limn→∞sn(8.5)3. a. 3pt Which of the following already has the form of a Fourier series on the interval [−2, 2].Explain!f(x) = 4 sin(π3x) g(x) =14+ 2 cos(3πx)b. 3pt We can find a sine series for the function f(x) = x3on the interval [0, 2]. To whatvalue does the sine series converge at x = 2? Explain!c. 3pt We can also find a cosine series for the function f(x) = x3on the interval [0, 2]. Towhat value does the cosine series converge at x = 2? Explain!d. 3pt Which one of the following partial differential equations can be solved by using thetechnique of separation of variables?ut= ux+ 1 ut+ ux= uExplain!4. 14pt Consider the functiong(x) =x if 0 ≤ x < 10 if 1 ≤ x ≤ 3Find a cosine series for the function g(x) on the interval [0, 2]. (Either use summation notationto write the answer or write the first four terms.)5. 12pt Determine all POSITIVE eigenvalues λ and the corresponding eigenfunctions for the followingtwo point boundary value problem:y00+ λy = 0 y0(0) = 0, y(3) = 06. Supp ose a thin homogeneous rod 5 cm long is insulated along its sides and m ade of a materialwith thermal diffusivity α2= 0.8 and that the left end is held at 10oand the right end is heldat 60o.a. 2pt What is the steady state solution to the above problem?b. 8pt It the initial temperature of the above rod is 60othen find the temperature u(x, t) ofthe rod at any time t > 0 and at any point x inside the rod 0 < x < 5. (If the answer involvesfinding a sine or cos ine series then DO NOT find the ac tual values of the anand/or bnwhichappear in the answer but indicate clearly what integrals must be evaluated to find them.)c. 4pt For the same problem as in Part b., approximately what is the temperature of the rodat x = 3 cm, after a long time.7. a. 6pt The displacement u(x, t) of a string of length 5cm with ends clamped satisfies thedifferential equation 9uxx= utt. Suppose that the initial displacement of this string is sin(5πx)and the initial velocity of the string is 0. Write down a formula for the displacement u(x, t) ofthe string at t > 0.(Hint: sin(5πx) is already in the form of a Fourier series.)a. 6pt Consider two identical thin rods having length 6cm and which are insulated exceptperhaps at the their ends. Also suppose that their temperatures u(x, t) satisfy the followingboundary conditions.Iu(x, 0) = 30 for 0 ≤ x ≤ 6u(0, t) = 0 = u(6, t) for t > 0IIu(x, 0) = 30 for 0 ≤ x ≤ 6ux(0, t) = 0 = ux(6, t) for t > 0Determine which will be warmer after a long time. Explain.8. a. 8pt 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) = f (y) if 0 < y < 7(Note: Your answer requires a sine or cosine series for f (y). Since no formula for f (y) is given,write but do not try to evaluate the formula for anor bn.)b. 8pt 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 w ith the values on the boundary given byF (θ). That is, find a function in polar coordinates u(r, θ), which is a solution of Laplace’sequation when r < 1 and which satisfies the following when r = 1:u(1, θ ) = F (θ) for − π < θ < π(Hint: F (θ) already has the form of a Fourier series on [−π, π].9. a. 10 pt Without using Laplace transforms, solve the following IVP:ty0= 2y + t3, y0(1) = 3b. 10pt Assume that f (t) is a piecewise continuous function with Laplace transformL{f(t)} = F (s). Derive the following formula for the Laplace transform of eatf(t):L{eatf(t)} = F (s − a)10. 14pt Without using Laplace transforms, solve the following IVP:y00+ 2y0− 3y = t , y(0) =79, y0(0)
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