# Write the system of equations in the image in matrix form x'_1(t)=3x_1(t)-2x_2(t)+e^tx_3(t) x'_2(t)=sin(t)x_1(t)+cos(t)x_3(t) x'_3(t)=t^2x_1(t)+tx^2(t)+x_3(t)

Question
Equations
Write the system of equations in the image in matrix form
$$\displaystyle{x}'_{{1}}{\left({t}\right)}={3}{x}_{{1}}{\left({t}\right)}-{2}{x}_{{2}}{\left({t}\right)}+{e}^{{t}}{x}_{{3}}{\left({t}\right)}$$
$$\displaystyle{x}'_{{2}}{\left({t}\right)}={\sin{{\left({t}\right)}}}{x}_{{1}}{\left({t}\right)}+{\cos{{\left({t}\right)}}}{x}_{{3}}{\left({t}\right)}$$
$$\displaystyle{x}'_{{3}}{\left({t}\right)}={t}^{{2}}{x}_{{1}}{\left({t}\right)}+{t}{x}^{{2}}{\left({t}\right)}+{x}_{{3}}{\left({t}\right)}$$

2020-11-18

Step 1
Now , we need to write this system of equations in matrix form .
The given system of equations is ,
$$\displaystyle{x}'_{{1}}{\left({t}\right)}={3}{x}_{{1}}{\left({t}\right)}-{2}{x}_{{2}}{\left({t}\right)}+{e}^{{t}}{x}_{{3}}{\left({t}\right)}$$
$$\displaystyle{x}'_{{2}}{\left({t}\right)}={\sin{{\left({t}\right)}}}{x}_{{1}}{\left({t}\right)}+{\cos{{\left({t}\right)}}}{x}_{{3}}{\left({t}\right)}$$
$$\displaystyle{x}'_{{3}}{\left({t}\right)}={t}^{{2}}{x}_{{1}}{\left({t}\right)}+{t}{x}^{{2}}{\left({t}\right)}+{x}_{{3}}{\left({t}\right)}$$
This system of equations is in 3 variables , $$\displaystyle{x}_{{1}}{\left({t}\right)},{x}_{{2}}{\left({t}\right)},{x}_{{3}}{\left({t}\right)}$$
Therefore , the matrix formed will be a $$\displaystyle{3}\times{3}$$ matrix .
Step 2
We can write down the system of equations in the form $$X'(t)=AX(t)$$.
Hence , we get,
$$\displaystyle{\left[\begin{array}{c} {x}_{{1}}'{\left({t}\right)}\\{x}_{{2}}'{\left({t}\right)}\\{x}_{{3}}'{\left({t}\right)}\end{array}\right]}={\left[\begin{array}{ccc} {3}&-{2}&{e}^{{t}}\\{\sin{{\left({t}\right)}}}&{0}&{\cos{{\left({t}\right)}}}\\{t}^{{2}}&{t}&{1}\end{array}\right]}{\left[\begin{array}{c} {x}_{{1}}{\left({t}\right)}\\{x}_{{2}}{\left({t}\right)}\\{x}_{{3}}{\left({t}\right)}\end{array}\right]}$$
Results: The system of equations can be written in matrix form as, $$\displaystyle{\left[\begin{array}{c} {x}_{{1}}'{\left({t}\right)}\\{x}_{{2}}'{\left({t}\right)}\\{x}_{{3}}'{\left({t}\right)}\end{array}\right]}={\left[\begin{array}{ccc} {3}&-{2}&{e}^{{t}}\\{\sin{{\left({t}\right)}}}&{0}&{\cos{{\left({t}\right)}}}\\{t}^{{2}}&{t}&{1}\end{array}\right]}{\left[\begin{array}{c} {x}_{{1}}{\left({t}\right)}\\{x}_{{2}}{\left({t}\right)}\\{x}_{{3}}{\left({t}\right)}\end{array}\right]}$$

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Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
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$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
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Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
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