Question

Given the linear system of first-order differential equations: $y_1' = y_1 + y_2 + 2y_3$ $y_2' = 2y_1 + y_2$ $y_3' = y_1 + y_3$ a- (2 pts.) Write the system as a matrix equation and show the coefficient matrix b- (6 pts.) Find the e-values and e-vectors of the coefficient matrix c- (6 pts.) Solve the system and write the expressions of $y_1(t)$, $y_2(t)$ and $y_3(t)$ if $y_1(0) = 0$, $y_2(0) = 1$ and $y_3(0) = 0$ Work items a to c and type the final answers in the answer box. Write legibly to show all the steps to the solution and include it in the show your work file attached to question 1.

          Given the linear system of first-order differential equations:
$y_1' = y_1 + y_2 + 2y_3$
$y_2' = 2y_1 + y_2$
$y_3' = y_1 + y_3$
a- (2 pts.) Write the system as a matrix equation and show the coefficient matrix
b- (6 pts.) Find the e-values and e-vectors of the coefficient matrix
c- (6 pts.) Solve the system and write the expressions of $y_1(t)$, $y_2(t)$ and $y_3(t)$ if $y_1(0) = 0$, $y_2(0) = 1$ and $y_3(0) = 0$
Work items a to c and type the final answers in the answer box. Write legibly to show all the steps to the solution and include it in the show your work file attached to question 1.

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Given the linear system of first-order differential equations:
y1' = y1 + y2 + 2y3
y2' = 2y1 + y2
y3' = y1 + y3
a- (2 pts.) Write the system as a matrix equation and show the coefficient matrix
b- (6 pts.) Find the e-values and e-vectors of the coefficient matrix
c- (6 pts.) Solve the system and write the expressions of y1(t), y2(t) and y3(t) if y1(0) = 0, y2(0) = 1 and y3(0) = 0
Work items a to c and type the final answers in the answer box. Write legibly to show all the steps to the solution and include it in the show your work file attached to question 1.

Added by Esperanza H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

07/12/2023

Step 1

Step 1: Writing the system as a matrix equation and showing the coefficient matrix: The given system of differential equations can be written in matrix form as: \[ \begin{pmatrix} y1' \\ y2' \\ y3' \end{pmatrix} = \begin{pmatrix} 1 & 1 & 2 \\ 2 & 1 & 0 \\ 1 & 0 & Show more…

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Thanks for your feedback!

Please be clear on part C having a hard time knowing what to do. Thank you. Given the linear system of first-order differential equations: y1' = y1 + y2 + 2y3 y2' = 2y1 + y2 y3' = y1 + y3 a- (2 pts.) Write the system as a matrix equation and show the coefficient matrix. b- (6 pts.) Find the eigenvalues and eigenvectors of the coefficient matrix. c- (6 pts.) Solve the system and write the expressions of y1(t), y2(t), and y3(t) if y1(0) = 0, y2(0) = 1, and y3(0) = 0. Work items a to c and type the final answers in the answer box. Write legibly to show all the steps to the solution and include it in the show your work file attached to question 1.

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Transcript

00:01 Hello everyone, here it is x1 dash and x2 dash is given.

00:04 So the first part is we have to write in matrix vector form.

00:08 So which is nothing but we can write this as minus 4 minus 10 3 7 x1 x2.

00:16 Then the second part is we have to find this system is homogeneous.

00:24 So yes it is homogeneous because x1 vector x2 vector is nothing but minus 5 3 e to the power 2t and 2 1 e to the power t forms the fundamental set of solution.

00:51 So we have x1 power of t will be minus 5 e to the power 2t 3 e to the power 2t.

01:00 So this implies x1 vector of t will be minus 10 e to the power 2t 6 e to the power 2t.

01:08 So the next step we write matrix vector form of equation.

01:13 So this is the matrix vector form of equation.

01:16 The next part is we have to give the fundamental matrix.

01:23 So the fundamental matrix is nothing but pi of t equal to x1 of t comma x2 of t.

01:34 So this is nothing but minus 5 e to the power 2t 3 e to power 2t 2 e to the power t e to the power minus t.

01:43 So this is the fundamental matrix.

01:46 Then the d part is we have to write the general solution of equation.

01:53 So that is x bar of t equal to pi of t c bar that is minus 5 e to the power 2t 2 e to the power t 3 e to the power t minus et c1 c2...

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