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yuradex [85]
3 years ago
5

Based on the provided information about the characteristic roots and the right hand side function g(t), determine the appropriat

e form of a particular solution to be used with the undetermined coefficient method.
(a) r1=-2i; r2=2i g(t)=2sin(2t) + 3cos(2t)
(b) r1=r2=0; r3=1 g(t)= t^2 +2t + 3
Mathematics
1 answer:
Rufina [12.5K]3 years ago
6 0

Answer:

Yp = t[Asin(2t) + Acos(2t)]

Yp = t²[At² + Bt + C]

Step-by-step explanation:

The term "multiplicity" means when a given equation has a root at a given point is the multiplicity of that root.

(a) r1=-2i; r2=2i g(t)=2sin(2t) + 3cos(2t)

As you can notice the multiplicity of this equation is 1 since the roots r1 = 2i and r2 = 2i appear for only once.

The form of a particular solution will be

Yp = t[Asin(2t) + Acos(2t)]

where t is for multiplicity 1

(b) r1=r2=0; r3=1 g(t)= t² +2t + 3

As you can notice the multiplicity of this equation is 2 since the roots r1 = r2 = 0 appears 2 times.

The form of a particular solution will be

Yp = t²[At² + Bt + C]

where t² is for multiplicity 2

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a. 38.19m/s

b. 38.605m/s

c. 38.937m/s

d. 39.0117m/s

e. 39.01917m/s

Step-by-step explanation:

The average velocity is defined as the relationship between the displacement that a body made and the total time it took to perform it. Mathematically is given by the next formula:

v_a_v_g = \frac{\Delta x}{\Delta t} =\frac{x_f-x_i}{t_f-t_i}

Where:

x_f=Final\hspace{3}distance\hspace{3}traveled\\x_i=Initial\hspace{3}distance\hspace{3}traveled\\t_f=Final\hspace{3}time\hspace{3}interval\\t_i=Initial\hspace{3}time\hspace{3}interval

a. Let's find h(3) and h(4) using the data provided by the problem:

h(3)=44(3)-0.83(3^2)=124.53=x_i\\h(4)=44(4)-0.83(4^2)=162.72=x_f

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b. Let's find h(3.5) using the data provided by the problem:

h(3.5)=44(3.5)-0.83(3.5^2)=143.8325=x_f

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c. Let's find h(3.1) using the data provided by the problem:

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The average velocity over the interval [3, 3.1] is :

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h(3.1)=44(3.01)-0.83(3.01^2)=124.920117=x_f

The average velocity over the interval [3, 3.01] is :

v_a_v_g=\frac{124.920117-124.53}{3.01-3} =39.0117m/s

e. Let's find h(3.001) using the data provided by the problem:

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v_a_v_g=\frac{124.5690192-124.53}{3.001-3} =39.01917m/s

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