7. Use the method of Laplace transform to solve the initial value problemy" − (a + ß)y' + aßy = (a − v) (B − 7)ert,y(0) = 1+a+b,y' (0) = aa+bB+Ya = 3, B = 4, y = 5, a = -4, b = 3.ß

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Answered: 7. Use the method of Laplace transform…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2. Use the Laplace transform to solve the initial value problem (a) x' - 5x = e-2t, x(0) = 3 ne lettre rouba23 (b) x' + 3x = 3t+7, x(0) = 1 (c) x" + 3x + 2x = 12et, (d) r" - 4x = t, on x(0) = 2, x(0) = 1, x'(0) = 3 x'(0) = 2.
21. (a) Use Newton's method to find the root of the equation r – xª + 3x2 – 3x – 2 = 0 in the interval [1,2] correct to six decimal places. (b) Use Newton's method to find the root of the equation 3x4 – 8x3 +2 = 0 in the interval [2,3] correct to six decimal places. 22. A particle is moving with the given data. Find the position of the particle. (a) v (t) = 3t – R, s (0) = 1. (b) a (t) = 2 sint + 3 cos t, s (0) = 0, v (0) = 2. 1+t² >
4. problem. Use the method of Laplace transform to solve the following initial value _y" + 3y' + 2y = 1, y(0) = 0, y'(0) = 2
14. Use the Laplace transform method to find the solution of the initial-value problem y" − 3y + 2y = 2x+1, y(0) = 2, y'(0) = -1.
If x(t) <---> X(f) and y(t) <---> Y(f), using the Fourier table and Fourier transform pairs, determine the Fourier transform of the following expressions. Identify and list the properties that you have used in the solution. Hint: You are not supposed to solve manually.
5. Solve simultaneously x[k+ 1] = 2x[k] + y[k]; y[k + 1] = 2x[k] + 3y[k] where x[0] = 3 and y[0] = −2. by using z-transform:
Part 2. Consider an ODE of the form x²y" + axy' + by = 0 with given constants a and b and unknown solution y(x). Assuming that y(x) follows the form y = xm, perform the following tasks: 2A. Solve for y'(x) and y" (x) and transform the given ODE in terms of x, m, a, and b. 2B. Determine the characteristic equation of the ODE. For each solution listed below, determine the corresponding roots of the characteristic equation and derive the respective Cauchy-Euler ODE: -3 2C. y = x ³ +1 2D. y = (1 + Inx)x` 2E. y = x[cos(2lnx) + sin(2lnx)]
Let a + 0, b + 0, c # 0, a # 0, B#0 and y # 0 represent any real nonzero constants, such that c > 0, a # B, a y and B # y. Let m >1 represent any positive integer.
A particular solution of the ODE y(3)+2y"-9y'=9 is defined by Yp=-X O Yp=x? 0
t3y'''+3t2y''=1. Use Variation of Parameters to solve this problem. Three solutions to the homogenous equatoin associated with this is (1/t), 1, and t
8. Use the method of Laplace transform to solve the initial value problem y" - 2ay' + a²y = (a - B)² est, y (0) = 1 + a, y'(0) = aa + B + b. = a = 1, b = 2, a = 10, 3 = 20.

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7. Use the method of Laplace transform to solve the initial value problem y" - (a + B)y' + aßy = (a −y)(B −y)ert, y(0) = 1+a+b, y'(0) = aa + b3 +7. a = 3,ß = 4, y = 5, a = -4, b = 3.