Question

please explain the steps as well! it’s imp for me to understand this question. i have attached the table for last part of the question

Consider the second order non-homogeneous constant coefficient linear ordinary differ- ential equation for y(x) ору , dy where Q(x) is a given function of r For each of the following choices of Q(x) write down the simplest choice for the particular solution yp(x) of the ODE. Your guess for yp(x) will involve some free parameters and should be sufficiently general to allow for a particular solution of the ODE. You are not required to solve the ODE for yp(x). Note: For this question your are only required to state the function yp(x) and thus you are not required to show any working (i) Q(x)- sin(2r) (ii) Q(x)-3x cos(3z +1) (iv) Q(x) = cosh(3x-2) (ii) Q(x) -e5* Derive the general solution of the ODE using Q(x) 3cos(2x). You should use the method of writing y(x) -yn(x) +yp(x). Be sure to clearly identify the ODE associated with yh Using your result from the previous question, find the particular solution of the ODE that satisfies the initial conditions y-1 and dy/dr 0 at x-0 Use a Laplace transform method to solve the ODE with Q(x)3cos(2r) as per question 2 and initial conditions as per question 3. Note: For this question you should use the table in Chapter 30 of the lecture notes to compute the Laplace transform and its inverse f(t) for s > 0 for s > a sinh(A) カfor s > AI cosh(Xt) for s>A sin(at) cos(ut) t for n2 for s > 0 +12 for s > 0 ta for α >-1 (t-a) F(s - a) f(t) lt-a) f(t-a) F(s) for 0 sF(s)-fo)

Answer 1

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