Kindly provide a detailed and thorough explanation about this example figure. It is only one example in two pictures. Determine the general solution to (D² + 4)(f) = sin(2x) satisfying the initial condition:f (0) = 0,D(f(0)) = 0Solution:The related homogeneous ODE is (D² + 4)(f) = 0 with characteristic polynomial r2 + 4 = 0. The rootsare +2i (multiplicity 1). Thus, the general solution to the homogeneous solution isuc(x) = c2 cos(2x) + c1 sin(2x)Finding an annihilator for sin 2x, A(D) = (D - 2i)(D + 2i) = D² + 4. Thus, a solution to the originalequation also satisfies(D² + 4)(D² + 4)f) = (D² + 4)(sin(2x))(D² + 4)²(f) = 0The characteristic polynomial is (r² + 4)² with root +2i (multiplicity 2). The corresponding solutions aree0(x) cos(2x) = cos(2x); e0(x) sin(2x) = sin(2x)xe0(x) cos(2x) = x cos(2x); xeº(x) sin(2x) = x sin(2x)±2i Hence, our particular solution is of the formu,(x) = c4x cos(2x) + c3x sin(2x) + C2 cos(2x) + c, sin(2x)which can be further simplified intoup(x) = c4x cos(2x)+ c3x sin(2x)Determining the values of the constants,(D² + 4)(f) = sin(2x)(D² + 4)(c4x cos(2x) + c3x sin(2x)) = sin(2x)D²(c4x cos(2x) + c3x sin(2x)) + 4(c4x cos(2x) + c3x sin(2x)) = sin(2x)D(-2c,x sin(2x) + C4 cos(2x) + 2c3x cos(2x) + cz sin(2x)) + 4c,x cos(2x) + 4c3x sin(2x) = sin(2x)-4c4 sin(2x) + 4c3 cos(2x) = sin(2x)We see that 4c3 = 0 and -4c, = 1. Hence,%3DC4 =Therefore, u, (x) = -x cos(2x) and the complete solution is of the form1-x cos(2x) + c2 cos(2x) + c, sin(2x)We use the initial conditions to determine c2 and c . Using the first initial condition,f(0) = c2 cos(0) + c; sin(0) = c2 = 0Using the second initial condition and substituting the value of c2,1DfD(-x cos(2x) + c2 cos(2x) + c; sin(2x))%3D= D: cos(2x) + c, sin(2x)1=x sin(2x) –cos(2x) + 2c, cos(2x)11D(f(0)) = -cos(0) + 2c, cos(0) = -i+ 2c, = 0Hence, c, = Therefore, the complete solution to the problem is11f(x) = -x cos(2x) + sin(2x)

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Answered: Determine the general solution to (D² +…

Determine the general solution to (D² + 4)(f) = sin(2x) satisfying the initial condition: f (0) = 0, D(f(0)) = 0 %3D Solution:

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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