Question 2 (a) Evaluate {e}b) EvaluateC{(t−2)² + sin 6t + cos 7t}c) Use the Laplace transform to solve the above initial value problemy" - 16y=e, y(0) = 1, y'(0) = 4.

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Answered: Question 2 ( a) Evaluate L{eot} 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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5.(i) Solve the differential equation (Dª – 1)y = 2 cos² x (ii) Find the Laplace transform of L[e 2t sin? t
A virus is spreading through the population of a salmon farm. It is determined that the growth rate of the virus varies with time of day according to the rule r(t) =(1+ sin(2nt)), where t is measured in days. The farm's total population is 80,000 salmon, and when the virus is first identified, the size of the infected population is lo = 50. %3D A.) Assuming that all of the fish are susceptible to the virus, which of the following equations best models the size of the infected population I(t) at time t: dI (i) dt (1+ sin(2nt)) · I dI 80000 – I (ii) (1+ sin(2nt)) ·I. dt 80000 dI I – 80000 - (iii) (1+ sin(2nt) · I · dt 80000 B.) Using the equation you chose and the available data, find I(t)
P2 Q1 (a)
(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
3. Using known DTFT's or results derived in class, sketch the signals y(t) whose Fourier series coefficients are given by 2 sin(kw/2) cos(kw/6) Yk kπ Use a fundamental period of To = 6 seconds.
2a
2 please show work
Let f(x, y) = xy² and r(t) = (¹/1²,t³). Calculate Vf. (Use symbolic notation and fractions where needed. Express numbers in exact form. Give your answer in the form (*, *). ) Vf= Calculate r' (t). (Use symbolic notation and fractions where needed. Express numbers in exact form. Give your answer in the form (*, *). ) r' (t) = Use the Chain Rule for Paths to evaluateƒ(r(t)) at t = 1.5. (Use decimal notation. Give your answer to three decimal places.) d dt - ƒ(r(1.5)) = = Use the Chain Rule for Paths to evaluate ƒ(r(t)) at t = -1.7. (Use decimal notation. Give your answer to three decimal places.) d dt d f(r(-1.7)) =
Use the formula {tf(t))} (s) = (-1)^({f}(s)) to help determine the following the expressions. dn ds" (a) {t cos bt} (b) {t² cos bt} Click here to view the table of Laplace transforms. (a) At cos bt}(s) =
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Question 2 ( a) Evaluate L{eot} b) Evaluate C{(t−2)² + sin 6t + cos 7t} c) Use the Laplace transform to solve the above initial value problem y" - 16y=e", y(0) = 1, y'(0) = 4.