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1 month ago

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Consider the vibrating system described by the initial value problem. (A computer algebra system is recommended.) u'' + u = 8 co

s ωt, u(0) = 5, u'(0) = 7 (a) Find the solution for ω ≠ 1.

2 answers:

Svet_ta [12.7K]1 month ago

5 0

Step-by-step explanation:

We are dealing with the differential equation

$u'' +u=8cos(\omega t)$

This can be approached using the method of undetermined coefficients. The overall solution is

u(t) = uh(t) + up(t)

where uh(t) represents the solution to the homogeneous part, and up(t) is a particular solution.

1. To start, let's find uh(t) using the characteristic polynomial:

$u'' +u=0$

$r^{2}+1=0$

The roots are imaginary:

r1=+i

r2=-i

Thus, uh(t) can be expressed as

$u_{h}(t)=Acos(t)+Bsin(t)$

where A and B are constants that need to be determined.

2. Next, we establish up(t). Using the method of undetermined coefficients, we can guess that up(t)=Ccos(wt)+Dsin(wt). Taking its derivatives gives us

$u_{p}'(t)=-\omega Csin(\omega t)+\omega Dcos(\omega t)\\u_{p}''(t)=-\omega ^{2}Ccos(\omega t)-\omega ^{2}Dsin(\omega t)$

Substituting this into the differential equation yields

$-\omega ^{2}Ccos(\omega t)-\omega ^{2}Dsin(\omega t)+Ccos(\omega t)+Dsin(\omega t)=8cos(\omega t)\\(-\omega ^{2}C +C)cos(\omega t) + (-\omega ^{2}D+D)sin(\omega t)=8cos(\omega t)$

From here, we obtain

$-\omega^{2}C+C=8\\-\omega^{2}D+D=0\\C=\frac{8}{1-\omega^{2}}\\D=0$

As it turns out, we set D=0 since the solution does not concern sin functions.

$u_{p}(t)=\frac{8}{1-\omega^{2}}cos(\omega t)$

3. Ultimately, the expression for u(t) is

$u(t)=Acos(t)+Bsin(t) +\frac{8}{1-\omega^{2}}cos(\omega t)$

When we apply the initial conditions, we find

$u(0)=A+\frac{8}{1-\omega^{2}} = 5\\u'(0)=B = 7\\B = 7\\A = 5-\frac{8}{1-\omega^{2}}$

PIT_PIT [12.4K]1 month ago

4 0

Answer:

u(t) = -(3 + w^2 ) cos t /(1- w^2)cos t + 7 sin t + 8 cos wt /(1- w^2)

Step-by-step explanation:

The characteristic equation is k² + 1 = 0, which leads to k² = -1, resulting in k = ±i.

The roots are k = i or -i.

The general solution takes the form u(x)=C₁cosx+C₂sinx.

Applying the method of undetermined coefficients, we have

Uc(t) = Pcos wt + Qsin wt

Calculating the derivatives gives us Uc’(t) = -Pwsin wt + Qwcos wt

And differentiating again yields Uc’’(t) = -Pw^2cos wt - Qw^2sin wt

With the equation U’’ + u = 8cos wt, we substitute:

-Pw^2cos wt - Qw^2sin wt + Pcos wt + Qsin wt = 8cos wt.

This simplifies to (-Pw^2 + P) cos wt + (-Qw^2 + Q) sin wt = 8cos wt.

From -Pw^2 + P = 8, we find P= 8 /(1- w^2).

From -Qw^2 + Q = 8, we can conclude Q = 0.

Thus, Uc(t) = Pcos wt + Qsin wt = 8 cos wt /(1- w^2).

Combining gives us U(t) = uh(t ) + Uc(t)

= C1cos t + c2 sin t + 8 cos wt /(1- w^2).

Initial conditions yield:

U(0) = C1cos(0) + c2 sin (0) + 8 cos (0) /(1- w^2)

Which leads us to C1 + 8 /(1- w^2) = 5

So C1 = 5 - 8 /(1- w^2) = -(3 + w^2 ) /(1- w^2).

Next, taking the derivative:

U’(t) = -C1 sin t + c2 cos t - 8 w sin wt /(1- w^2).

Evaluating at t = 0 gives us:

U’(0) = -C1 sin (0) + c2 cos (0) - 8 w sin (0) /(1- w^2) = 7.

Thus, c2 = 7.

u(t) = -(3 + w^2 ) cos t /(1- w^2)cos t + 7 sin t + 8 cos wt /(1- w^2)

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Answer:

a) Slope: $\hat \beta_1 =\frac{7625.9}{1248.9}=6.106$

Intercept: $\hat \beta_o = 157.955 -6.106 (69.686)=-267.548$

b) $r=\frac{7625.9}{\sqrt{[1248.9][94228.8]}}=0.657$

Additionally, the coefficient of determination is $r^2 = 0.657^2 =0.432$

Step-by-step explanation:

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The correlation coefficient is a measure that quantifies the strength of the relationship between two variable movements, denoted as r and ranging between -1 and 1.

The sum of squares refers to the total of the squared deviations, where deviation is defined as the difference between each value and the grand mean.

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n=110, $\sum x_i y_i = \sum (X-\bar X)(Y-\bar Y) =7625.9,\sum x^2_i=\sum (x-\bar x)^2 =1248.9, sum y^2_i=\sum(y-\bar y)^2 =94228.8$

$\sum Y_i =17375, \sum X_i = 7665.5$

Part a

The slope can be calculated using this formula:

$\hat \beta_1 =\frac{\sum (x-\bar x) (y-\bar y)}{\sum (x-\bar x )^2}$

Following the substitutions, we have:

$\hat \beta_1 =\frac{7625.9}{1248.9}=6.106$

The intercept can be determined with this formula:

$\hat \beta_o = \bar y -\hat \beta_1 \bar x$

Average values for x and y can be calculated this way:

$\bar X=7665.5/110 =69.686, \bar y= 17375/110=157.955$

Replacing yields:

$\hat \beta_o = 157.955 -6.106 (69.686)=-267.548$

Part b

The correlation coefficient can be calculated using the following formula:

$r=\frac{\sum (x-\bar x)(y-\bar y) }{\sqrt{[\sum (x-\bar x)^2][\sum(y-\bar y)^2]}}$

In our situation:

n=110, $\sum x_i y_i = \sum (X-\bar X)(Y-\bar Y) =7625.9,\sum x^2_i=\sum (x-\bar x)^2 =1248.9, sum y^2_i=\sum(y-\bar y)^2 =94228.8$

We can compute the correlation coefficient by substituting values:

$r=\frac{7625.9}{\sqrt{[1248.9][94228.8]}}=0.657$

The coefficient of determination is $r^2 = 0.657^2 =0.432$

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Answer:

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Joaquin is four more than half Marjorie’s age. Marjorie is also 32 less than three times Joaquin’s age.

babunello [11817]

Answer:

Her age is 40 years

Step-by-step explanation:

Let Jerry's age be represented as j

Let Marjorie's age be denoted as m

The first part of the question states:

j = 4 + m/2

For the second equation:

m = 3j - 32

Multiply the first equation by 2 to get:

2j = 8 + m

Leading to m = 2j - 8

Set both equations for m equal to each other:

2j - 8 = 3j - 32

By simplifying, we get:

3j - 2j = 32 - 8

Therefore, j = 24

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m = 2(24) - 8

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Answer:

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The probability can be determined using the complement rule, with the standard normal distribution, an excel sheet, or a calculator.

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b) $P(X$

This probability can also be calculated using the normal standard distribution, an excel sheet, or a calculator.

$P(z$

c) $P($

For this one, the probability can likewise be derived from the standard normal distribution, excel, or a calculator, with specific adjustments:

$P(-1.43$

Step-by-step explanation:

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Normal distribution refers to a symmetric probability distribution centered around the mean, indicating that occurrences near the mean are more common than those far from it.

The Z-score is a statistic that represents a value's relationship to the average of a set of values, expressed in terms of how many standard deviations it is away from the mean.

Part a

Let X denote the random variable representing the lengths within a population, and for our case, the distribution for X is as follows:

$X \sim N(15,3.5)$

Where $\mu=15$ and $\sigma=3.5$

We seek the probability:

$P(X>20)$

The most effective way to solve this is by leveraging the normal distribution and the corresponding Z-score:

$z=\frac{x-\mu}{\sigma}$

By applying this formula, we can find the probability:

$P(X>20)=P(\frac{X-\mu}{\sigma}>\frac{20-\mu}{\sigma})=P(Z>\frac{20-15}{3.5})=P(z>1.43)$

Again, this probability can be obtained either using the complement rule, the standard normal distribution, or a calculator.

$P(z>1.43)=1-P(z$

Part b

$P(X$

This probability can also be computed using either the normal standard distribution, an excel sheet, or a calculator.

$P(z$

Part c

$P($

In this case, the probability can similarly be acquired with the help of the standard normal distribution, an excel sheet, or a calculator, with particular adjustments:

$P(-1.43$

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Response: 7

Detailed explanation:

A Venn diagram can help visualize this problem.

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Moreover, there are 5 students who wish to study only Latin.

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In the same manner, those wishing to study only French amount to 16 - 4 - 3 - 2 = 7.

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