1. (LT-2) Use the Laplace transform to solve the initial value problem y" + y = f(t),y(0) = v'(0) = 0, in which f(t) defined piecewise as follows:0 0≤x≤T4 x < 2π02п < xWrite your solution as a piecewise-defined function.

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Answered: 1. (LT-2) Use the Laplace transform to…

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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3 (5) (10 pts.) Let a function y = f(x) be implicitly given by the equation y² - x²y³ - cos (Ty) = xª. Find f'(x) at the point (-1, 1).
(a) Faraday's law characterizes the voltage drop across an inductor such as di V = L- dt where V, = voltage drop (V), L = inductance (in henrys; 1 H = 1 V s/A), i = current (A) and t = time (s). Suppose that the current through the inductor is represented by the function such as i(t) = (20 – t)? + (20 – t) cos(VT). i. Use the centered difference formula, O(h*) to estimate the voltage drop at t = 10 s for an inductance of 3 H using a step size, h = 3 accurate to 3 decimal places.
(b) Find the derivative of f (1) = (12 +6) (4z – 9) by using the product rule. Let g (z) = 12 + 6 %3D and h (z) = 4z – 9. ya sin (a) la sin (a) la f (z) =
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.
(b) Suppose that the function f (x) has the form f(x) : тх + b' where C is a non-zero constant and m + 0. Explain why it is impossible for f(x) to have any local max's or local min's.
Suppose f(x) is differentiable at x = 2 and that f(2) = 3 and f′(2) = −4.
Find the derivative of following functions. 1 g(x) = Vr? – 1 sec-1x 7 y = (2r + 1) (x* – 3)6 %3D 2 F(0) = arcsin Vsin 0 sin? r tan a (x2 + 1)2 8 y = 3 y = tan-(x + V1+x2) %3D x2 + 1 4 F(y) = y ln(1 + e®) 9 y = x2 – 1 | 5 y = In(e + xe") p2 + log7(r) 10 g(r) = 7" 6 y = (tan r)
Q1(2)
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)]
Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function -4-5(t-1), (0)-6, (0) 0. = a Find the Laplace transform of the solution Y()-C (v(t)}- b. Obtain the solution y(t). (1) c Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at 1. if 0 < t < 1, -{₁ u(t)= if 1≤ t < 00.
* 1 – t² dt 1+ t4 Differentiate f(x) = 1

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1. (LT-2) Use the Laplace transform to solve the initial value problem y" + y = f(t), y(0) = y'(0) = 0, in which f(t) defined piecewise as follows: 0 0