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)

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
asked 2020-11-17
Write the system of equations in the image in matrix form
\(\displaystyle{x}'_{{1}}{\left({t}\right)}={3}{x}_{{1}}{\left({t}\right)}-{2}{x}_{{2}}{\left({t}\right)}+{e}^{{t}}{x}_{{3}}{\left({t}\right)}\)
\(\displaystyle{x}'_{{2}}{\left({t}\right)}={\sin{{\left({t}\right)}}}{x}_{{1}}{\left({t}\right)}+{\cos{{\left({t}\right)}}}{x}_{{3}}{\left({t}\right)}\)
\(\displaystyle{x}'_{{3}}{\left({t}\right)}={t}^{{2}}{x}_{{1}}{\left({t}\right)}+{t}{x}^{{2}}{\left({t}\right)}+{x}_{{3}}{\left({t}\right)}\)

Answers (1)

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, \(\displaystyle{\left[\begin{array}{c} {x}_{{1}}'{\left({t}\right)}\\{x}_{{2}}'{\left({t}\right)}\\{x}_{{3}}'{\left({t}\right)}\end{array}\right]}={\left[\begin{array}{ccc} {3}&-{2}&{e}^{{t}}\\{\sin{{\left({t}\right)}}}&{0}&{\cos{{\left({t}\right)}}}\\{t}^{{2}}&{t}&{1}\end{array}\right]}{\left[\begin{array}{c} {x}_{{1}}{\left({t}\right)}\\{x}_{{2}}{\left({t}\right)}\\{x}_{{3}}{\left({t}\right)}\end{array}\right]}\)

0

Relevant Questions

asked 2021-01-13
Solve the following system of equations.
\(\displaystyle{2}{x}_{{1}}−{x}_{{2}}−{x}_{{3}}=−{3}\)
\(\displaystyle{3}{x}_{{1}}+{2}{x}_{{2}}+{x}_{{3}}={13}\)
\(\displaystyle{x}_{{1}}+{2}{x}_{{2}}+{2}{x}_{{3}}={11}\)
\(\displaystyle{\left({x}_{{1}},{x}_{{2}},{x}_{{3}}\right)}=\)
asked 2021-02-09
Solve the following system of equations. (Write your answers as a comma-separated list. If there are infinitely many solutions, write a parametric solution using t and or s. If there is no solution, write NONE.)
\(\displaystyle{x}_{{1}}+{2}{x}_{{2}}+{6}{x}_{{3}}={6}\)
\(\displaystyle{x}_{{1}}+{x}_{{2}}+{3}{x}_{{3}}={3}\)
\(\displaystyle{\left({x}_{{1}},{x}_{{2}},{x}_{{3}}\right)}=\)?
asked 2020-12-17
Write the vector form of the general solution of the given system of linear equations.
\(3x_1+x_2-x_3+x_4=0\)
\(2x_1+2x_2+4x_3-6x_4=0\)
\(2x_1+x_2+3x_3-x_4=0\)
asked 2021-05-07
Solve the following differential equation by using linear equations.
\(dx/dt = 1- t + x - tx\)
asked 2021-01-16
Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number \(c_1\). \(c_1\cdot\begin{cases}x_1+x_2+x_3=0\\5x_1-2x_2+2x_3=0\\8x_1+x_2+5x_3=0\end{cases},\ X=c_1\begin{pmatrix}4\\3\\-7\end{pmatrix}\)
asked 2020-11-09
Write the system of linear equations in the form Ax = b and solve this matrix equation for x.
\(\begin{cases}x_1+x_2-3x_3=-1\\-x_1+2x_2=1\\x_1-x_2+x_3=2\end{cases}\)
asked 2021-03-22
Write the parametric equations
x=3t-1, y=3-3t
as a function of x in the Cartesian form.
y=
asked 2021-02-21
Solve the given system of equations by matrix equation.
5x-4y=4
3x-2y=3
asked 2021-05-05

A random sample of \( n_1 = 14 \) winter days in Denver gave a sample mean pollution index \( x_1 = 43 \).
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.
\( H_0:\mu_1=\mu_2.\mu_1>\mu_2 \)
\( H_0:\mu_1<\mu_2.\mu_1=\mu_2 \)
\( H_0:\mu_1=\mu_2.\mu_1<\mu_2 \)
\( H_0:\mu_1=\mu_2.\mu_1\neq\mu_2 \)
(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.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
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.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
\( \mu_1 - \mu_2 \).
(Round your answers to two decimal places.)
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
asked 2020-12-28
Write the vector form of the general solution of the given system of linear equations.
\(x_1+2x_2-x_3=0\)
\(x_1+x_2+x_3=0\)
\(x_1+3x_2-3x_3=0\)
...