part 2)a) Solvey" - 2y' +4y= 0.b) Write down the complementary solution ofy"-2y' +4y=1+tc) Use the Laplace transform to solve the giveninitial value problemy"-2y' +4y=1+t, y(0)=0, y'(0) = 2

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Answered: part 2) a) Solve y" - 2y' +4y= 0. b)…

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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Determine the solutions of the following first-order ODEs: Tip:y'=xv'+v y=xv * xv'+v=(4xv-3x)/(2x-xv) >>>> xv'+v=(2v-3)/(1-v)>>>xv'=(2v-3-v+v²)/(1-v) * >>>>>>>>>** ANSWER CORRECT :|y − x| = C|y + x|3 **<<<<<< *****The answer |y − x| = C|y + 3x|5 is considered wrong ( will not be accepted as it will be considered wrong)*****
(5) Find the transfer function h(@)= y(@)/x(@) of the following ODE and plot its magnitude. y"(t)+y'(t) + y(t) = x(t) HINT: Take the Fourier Transform of the ODE to obtain - a²ŷ(0) + iaỹ(a) + y(@) = x(w) Giving h(o)= = y(@) 1 x(@) - @²+ io +1 The magnitude of the transfer function is then |ħ(w) = 1 1- @²} + @² |h(@) 0.5 0.8 0.7 0.6 0.5 0.4 0.3 e 0.2 0,4 0.6 0.8 (10) e 1 1.2 1.4 1.6 1.8 2
Question 9 Find the solution of the integro-differential equation using Laplace Transform: y" + 4y' + 3y = 2 e-tdt;y(0) = 0,y'(0) = 0 %3D Oy = (4 – 3e(1+ 2t) – e-3*] Oy = (4+ 3e-(1– 2) – e-3«] Oy = (4+ 3e (1 – 2t) – e*] Oy = }(4 – 3eʻ (1 + 2t) – e*] %3D
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)]
2
The characteristic equation of the ODE y(3) -2y'+y?=0 is defined by p3 – 2 =0 p – 2r+1=0 r³-2r=0 3-2r²+r=0
part 2) a) Solve y" -6y' +9y=0. b) Find the particular solution of y"-6y' +9y=3-t c) Use the Laplace transform to solve the given initial value problem y"-6y' +9y=3-t, y(0)=0, y' (0) = 2

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part 2) a) Solve y" - 2y' +4y= 0. b) Write down the complementary solution of y"-2y' +4y=1+t c) Use the Laplace transform to solve the given initial value problem y"-2y' +4y=1+t, y(0)=0, y'(0) = 2