midterm3sol - Solutions to the Second Midterm Exam Problem...

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Unformatted text preview: Solutions to the Second Midterm Exam Problem 1. a) We look for a solution in the form y = ae 2 x for some a . Substituting y into the equation, we get 4 ae 2 x + 2 ae 2 x + ae 2 x = e 2 x , from which 7 a = 1 and a = 1 / 7. Answer: A particular solution is y = e 2 x 7 . b) First, we look for a particular solution in the form y = ae 2 x for some a . Thus we get y = 4 ae 2 x and the equation becomes 4 ae 2 x- 4 ae 2 x = e 2 x , which is a signature equation of duplication (resonance). Hence we try y = axe 2 x instead, so y = e 2 x ( a + 2 ax ) and y = e 2 x (2 a + 4 ax ) + 2 ae 2 x = e 2 x (4 a + 4 ax ). Substituting that into the equation, we get e 2 x (4 a + 4 ax )- 4 axe 2 x = 4 ae 2 x = e 2 x , from which a = 1 / 4. Answer: A particular solution is y = xe 2 x 4 . c) First, we solve the homogeneous equation y - y = 0 . Looking for the solution in the form y = e rx we get r 2- 1 = 0, from which the general solution is y = C 1 e x + C 2 e x for some constants C 1 and C 2 . Now, we look for the solution of the original equation in the form y = C 1 ( x ) e x + C 2 ( x ) e x . Differentiating, we get y = C 1 e x + C 1 e x + C 2 e x- C 2 e x , 1 which, after the imposing the condition (1.1) C 1 e x + C 2 e x = 0 becomes just y = C 1 e x- C 2 e x . Differentiating further, we get y = C 1 e x + C 1 e x- C 2 e x + C 2 e x . Hence (1.2) y - y = C 1 e x- C 2 e x = xe 2 x . Now we solve (1.1) and (1.2) for C 1 and C 2 . If we add (1.1) and (1.2), we get 2 C 1 e x = xe 2 x , so C 1 = xe x / 2. Similarly, if we subtract (1.2) from (1.1), we get 2 C 2 e x =- xe 2 x , so C 2 =- xe 3 x / 2. Integrating by parts, we get C 1 ( x ) = 1 2 integraldisplay xe x dx = xe x 2- e x 2 + constant and C 2 ( x ) =- 1 2 integraldisplay xe 3 x dx =- xe 3 x 6 + e 3 x 18 + some other constant. Since we are interested in particular solutions, we can choose the constants equal to 0 and get y = C 1 ( x ) e x + C 2 ( x ) e x = xe 2 x 2- e 2 x 2- xe 2 x 6 + e 2 x 18 = xe 2 x 3- 4 e 2 x 9 . Answer: A particular solution is y ( x ) = e 2 x parenleftbigg x 3- 4 9 parenrightbigg . d) From the general equation we have mx + kx = 0, where m = 10 and k is determined from the equation F = kx with F = 5 and x = 1, so k = 5. Thus the equation reads 10 x + 5 x = 0 or, equivalently, x + 1 2 x = 0, from which the general solution is x ( t ) = A sin t 2 + B cos t 2 . Hence the resonance occurs at = 1 / 2 and =- 1 / 2. Answer: The resonance occurs at = 1 / 2, and, if we assume that can be negative, at =- 1 / 2....
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midterm3sol - Solutions to the Second Midterm Exam Problem...

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