Math 308 Section 5 17 Exam 2 – Jean Marie Linhart Spring 2009“An Aggie does not lie, cheat, or steal or tolerate those who do”On my honor as an Aggie, I have neither given nor receivedunauthorized aid on this exam.Printed name:Signature:xkcd.com• You may use your 4x6 inch notecard for this exam. You must hand it inwith your exam.• You may not use any other notes, a calculator, or your book.• Your cellphone must be turned off and put away dur ing this exam!• You may not collaborate with your neighbors on this ex am.• You must show all appropriate work to receive credit, especially partialcredit.• If you use a formula, WRITE IT DOWN.• The instructor will provide additional scratch paper if needed.• Read each question carefully.• GOOD LUCK!!!!!!!121) (10 points) Find an explicit solution to the following:d2ydx2− 2dydx+ 3y = 0 y(0) = 0, y′(0) = −2√2Hint: Use the quadratic formula.32) (10 points) Find the genera l solution of the Cauchy-Euler equationx2y′′− 3xy′+ 4y = 0(10 points) Now use your work from above and find the general solution forx2y′′− 3xy′+ 4y = x243) (8 points) According to the existence and uniqueness theorem for 2 nd orderlinear ODEs, on what intervals might the following ODE have unique solutions?(x2− 3x)y′′+ 2xy′− y = x24) (5 points) YOU DO NOT HAVE TO SOLVE THIS. If you were usingthe method of undetermined coefficients to so lve this e quation, what form of aparticular solution would you a ttempt (guess) and why?y′′+ 2y′+ y = e−x5) (3 p oints) Use one step of the Euler method to calculate a n approximationto y(1.1) for the IVPy′= 2x2− xy, y(1) = 2(4 p oints) Use one step of the Improved Euler Method to c alculate an approxi-mation to y(1.1) for the IVP above.56) (18 points) Find the solution to the IVPd2ydx2− 5dydx+ 4y =343cos(x) y(0) = 1, y′(0) = 167) (15 points) Find the genera l solution toy′′+ 4y′+ 4y = e−2xln(x)78) (5 points) Show the Laplace transform of f (t) = sin(bt) isL{f }(s) =bs2+ b2by using the integral definition of the Laplace transform and perfor ming theintegral. You may use the integration formulaZecxsin(ax)dx =1a2+ c2ecx[c sin(ax) − a cos(ax)] + Crather than integrating by parts twice!(2 p oints) For what values of s does L{f }(s) exist for f (t) = sin(bt)?(5 points) Use the integral definition of Laplace transform to show that ifL{g(t)}(s) = G(s) thenL{eatg(t)}(s) = G(s − a)(5 p oints) What isL{7e3tsin(2t) − 3e4t}?You do not nee d to use the integral definition of the Laplac e transform, but youmust show your work!89) Extra credit. Choose a nd do at most one of the following.a. (5 points extra credit) The variation of parameters method assumes you havey1(x) and y2(x) which are linearly independent solutions to the homogeneousequationL[y] = y′′+ p(x)y′+ q(x)y = 0then in order to solveL[y] = y′′+ p(x)y′+ q(x)y = g(x)you attempt a solution of the form v1(x)y1(x) + v2(x)y2(x) = yp(x) with therestriction that v′1(x)y1(x) + v′2(x)y2(x) = 0. Using this and the differentialequation above, derive the two formulas for v1(x) and v2(x).orb. (3 points extra credit) Return to your solution to pro blem 1, which was tofind a solution to the IVPd2ydx2− 2dydx+ 3y = 0 y(0) = 0, y′(0) = −2√2Would it make sense for this e quation to represent a mass-spring oscilla tor inthe real world? Why or why no
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