Consider the following initial value problem:y" + 25y = f(t); _y(0) = 0, y'(0) = 5;f(t)y(t) ===7654325, 0≤t <3³5{(a) Sketch the graph of the forcing function on an appropriate interval.103π0, 3≤ t <∞5Ť(b) Find the solution of the given initial value problem.NOTE: Denote the Heaviside function by ue(t) where uc(t) = 1 if t > c and 0 otherwise.Indicate separately the value of c.3

Consider the following initial value problem:y" + 25y = f(t); y(0) = 0, y'(0) = 5;f(t):=5, 0 ≤ t…

Answered: Consider the following initial value…

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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Consider the following initial value problem: y" +9y = sint - U2π(t) sin(t – 2π); y(0) = 0, y'(0) = 0; (a) Sketch the graph of the forcing function on an appropriate interval J π/2 3л/4 5/4 3/2 7/4 2 1 (b) Find the solution of the given initial value problem. NOTE: Denote the Heaviside function by u(t) where ue(t) = 1 if t≥c and 0 otherwise Indicate separately the value of c. y(t) = = where c = (c) Use a graphing utility to view the graph of the solution. (d) Explain how the graphs of the forcing function and the solution are related. When the external force is on, the solution is periodic with a period and an amplitude 9/4 5/2 11n/43 Choose one ▾ When the external force is removed, the solution [Choose one ▾
F(t) Find the steady periodic solution Xp (t) = C cos (@t-a) of the given equation mx" + cx' + kx = F(t) with periodic forcing function F(t) of frequency w. Then graph Xsp (t) together with (for comparison) the adjusted function F₁ (t) = mo x" + 12x' + 11x = 11 cos 2t Xsp (t) = (Use integers or decimals for any numbers in the expression. Round to three decimal places as needed.) Choose the correct graph below. O A. O G B. Q C O C. O D. 18 HALD
4. (a) Consider the ODE y(1+r³)y = r². Determine where in the xy-plane existence and uniqueness issues to an associated initial value problem may occur. (b) Solve the IVP y(1+x³)y′ = x², y(0) = yo and determine how the interval in which the solution exists depends on the initial value yo.
1. Find the maximum value of the function f(x,y) = 4x + 2y+x² – 2x* + 2xy – 3y² on the interval -2 to 2 for both x and y employing the specified method below: (a) random search method using google sheet or excel with 500 iterations. (b) equating each partial derivative of fwith respect to x and partial derivative of f with respect to y to zero, and then solve the nonlinear system of equations using fixed-point iteration or Newton's Method. (c) random search method using a program that you created.
#4. Consider the function f (x, y) = x²y+ y3 at the point (1,-2). (a) Calculate the partial derivatives f, and f, and explain what property of those functions at the given point implies that f is differentiable at this point. (b) Find the linearization L(x, y) of this function at that point.
7. A scientist found that the amount Q(t) of a substance present at time t 20 satisfies the equation: 40"(t) + 4Q'(t) + Q(t) = 0 (a) Verify Q(t) = A(1+t) e-0.5t satisfies the equation for A> 0 (b) If Q(0) = 10mg, find the maximum value of Q(t) and the time at which it occurs What happens to the amount of Q(t) as t→∞? (c)
1. Find an interval for which the following has an unique solution: (x-2)y"+3y = x, y(3) =2,y'(3) = 1
8. Here's a whole new example function: f(x, y) = x? + 2xy + y³. fe (a) Calculate af (b) Calculate dy fe = 0 and af = 0. (Points where all partial derivatives dy (c) Solve the system of equations of a function are zero are again called critical points of the function.) (Did you find two critical points, (x, y) = (0, 0) and (x, y) = (-3, )?) 3/3
Consider the following initial value problem: y" + 4y = sint - u2(t) sin(t – 2π); y(0) = 0, y'(0) = 0; (a) Sketch the graph of the forcing function on an appropriate interval. 1 y(t): 4 m/2 3/4 m 5/4 3m/2 7/4 2 9/4 5/2 1/4 3 (b) Find the solution of the given initial value problem. NOTE: Denote the Heaviside function by ue(t) where ue(t) = 1 if t≥c and 0 otherwise. Indicate separately the value of c. = where c = X X X
y ″ + 4y = sint − u2π(t) sin(t − 2π) with initial condition y ( 0 ) = 0 and y ′(0) = 0 sketch the graph of the forcing function on an appropriate interval.
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Find the steady periodic solution Xsp (t) = C cos (wt - x) of the given equation mx'' + cx' + kx = F(t) with periodic forcing function F(t) of frequency w. Then graph Xsp (t) together with (for comparison) F(t) the adjusted function F₁ (t) = - x' + 6x' + 9x = 11 cos 2t то = Xsp (t) (Use integers or decimals for any numbers in the expression. Round to three decimal places as needed.)

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