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Consider the 2-dimensional system of linear equations -2 X' = 2 Note that the coefficient matrix...
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and ide...
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and identify the matrix A. Obtain a general solution to the matrix differential equation in terms of eigenvectors and eigenvalues of A. Justify your answer (b) Classify possible types and stability of the equilibrium with dependence on the eigenvalues of A. (Note: You are not asked to compute the...
Consider the following coefficient matrix, which contains a parameter a. 11 6 (a) Determine the e...
Consider the following coefficient matrix, which contains a parameter a. 11 6 (a) Determine the eigenvalues in terms of α. Supposing that α > 0, enter your answers in increasing order. Equation Editor Common Ω Matrix 自0 tania) sin(a) d a 4 secia) esia)a) costa) 邇 alal sin"(a) cos-1(a) tan-"(a)- u oo Ω Matrix cosa) tana) ,..tseela, osia, =a) Va ya lal sin-(a)(a) tan ( o sinia) sec(α) //u),dx ! 읊 cscla) (b) Find the critical value or values of...
Problems 17 through 19 deal with competitive systems much like those in Examples 1 and 2 except t...
#19 all parts
Problems 17 through 19 deal with competitive systems much like those in Examples 1 and 2 except that some coefficients depend on a parameter a. In each of these problems, assume that x, y, and a are always nonnegative. In each of Problems 17 through 19: (a) Sketch the nullclines in the first quadrant, as in Figure 9.4.5. For different ranges of a your sketch may resemble different parts of Figure 9.4.5 (b) Find the critical points...
Consider the linear system. dy da dt = + 2y, at 9x + 4y. (1). Find...
Consider the linear system. dy da dt = + 2y, at 9x + 4y. (1). Find the eigenvalues. (2). Find the eigenvectors. (3). Determine the type and stability of the critical point(0,0). (4). Roughly sketch the phase portrait, including directions.
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each...
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each of the regions b. Linearize the system around the equilibrium point, and use your result to classify the type of the c. Use the information from parts a and b to sketch the phase portrait of the system. 2. Sketch the phase portraits for the following systems...
2. Use the nonlinear system below to answer each part. 5.2 x = y-1, y =...
2. Use the nonlinear system below to answer each part. 5.2 x = y-1, y = -y + 1 -2 (a) Plot the nullclines and indicate the direction of the orbits in each region sepa- rated by the nullclines (do not use software, do this by hand). (b) Find all equilibria for the system. (c) Determine the Jacobian matrix for the system and use it to classify the local stability properties of each equilibrium in part (b).
Consider the 3-dimensional system of linear equations 1 1 1 X' = X 2 1 -1...
Consider the 3-dimensional system of linear equations 1 1 1 X' = X 2 1 -1 0-1 1 (a) Find a fundamental set of solutions for this system. Note that -1 is one of the eigenvalues (b) Find the general solution, and use it to find the solution satisfying -4 X(0) 2
Consider the 3-dimensional system of linear equations 1 1 1 X' = X 2 1 -1 0-1 1 (a) Find a fundamental set of solutions for this system....
Consider the system of differential equations ; 1) Find the fixed points of the system , 2) Eval...
consider the system of differential equations ; 1) Find the fixed points of the system , 2) Evaluate the Jacobian Matrix at each fixed point, 3) Classify stability of each fixed point, 4) Sketch the graph of the phase portrait,
Suppose 7' = AT, where A is the 2 x 2 matrix below. A= (1 1...
Suppose 7' = AT, where A is the 2 x 2 matrix below. A= (1 1 1 3 (a) Determine the eigenvalues and eigenvectors of A. (b) Express the general solution of t' = Az in terms of real valued functions. (c) Sketch the phase portrait of the system. Do not forget to label your axes.
4. Consider the system y'- Ay(t), for t > 0, with A - 1 -2 (a) Show that the matrix A has eigenva...
4. Consider the system y'- Ay(t), for t > 0, with A - 1 -2 (a) Show that the matrix A has eigenvalues וג --1and Az--3 with corresponding eigenvectors u (1,1) and u2 (1,-1) (b) Sketch the trajectory of the solution having initial vector y(0) = ul. (c) Sketch the trajectory of the solution having initial vector y(0) -u2. (d) Sketch the trajectory of the solution having initial vector y(0)-u -u 1 U
4. Consider the system y'- Ay(t), for...
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and identify the matrix A. Obtain a general solution to the matrix differential equation in terms of eigenvectors and eigenvalues of A. Justify your answer (b) Classify possible types and stability of the equilibrium with dependence on the eigenvalues of A. (Note: You are not asked to compute the...
Consider the following coefficient matrix, which contains a parameter a. 11 6 (a) Determine the eigenvalues in terms of α. Supposing that α > 0, enter your answers in increasing order. Equation Editor Common Ω Matrix 自0 tania) sin(a) d a 4 secia) esia)a) costa) 邇 alal sin"(a) cos-1(a) tan-"(a)- u oo Ω Matrix cosa) tana) ,..tseela, osia, =a) Va ya lal sin-(a)(a) tan ( o sinia) sec(α) //u),dx ! 읊 cscla) (b) Find the critical value or values of...
#19 all parts
Problems 17 through 19 deal with competitive systems much like those in Examples 1 and 2 except that some coefficients depend on a parameter a. In each of these problems, assume that x, y, and a are always nonnegative. In each of Problems 17 through 19: (a) Sketch the nullclines in the first quadrant, as in Figure 9.4.5. For different ranges of a your sketch may resemble different parts of Figure 9.4.5 (b) Find the critical points...
Consider the linear system. dy da dt = + 2y, at 9x + 4y. (1). Find the eigenvalues. (2). Find the eigenvectors. (3). Determine the type and stability of the critical point(0,0). (4). Roughly sketch the phase portrait, including directions.
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each of the regions b. Linearize the system around the equilibrium point, and use your result to classify the type of the c. Use the information from parts a and b to sketch the phase portrait of the system. 2. Sketch the phase portraits for the following systems...
2. Use the nonlinear system below to answer each part. 5.2 x = y-1, y = -y + 1 -2 (a) Plot the nullclines and indicate the direction of the orbits in each region sepa- rated by the nullclines (do not use software, do this by hand). (b) Find all equilibria for the system. (c) Determine the Jacobian matrix for the system and use it to classify the local stability properties of each equilibrium in part (b).
Consider the 3-dimensional system of linear equations 1 1 1 X' = X 2 1 -1 0-1 1 (a) Find a fundamental set of solutions for this system. Note that -1 is one of the eigenvalues (b) Find the general solution, and use it to find the solution satisfying -4 X(0) 2
Consider the 3-dimensional system of linear equations 1 1 1 X' = X 2 1 -1 0-1 1 (a) Find a fundamental set of solutions for this system....
Suppose 7' = AT, where A is the 2 x 2 matrix below. A= (1 1 1 3 (a) Determine the eigenvalues and eigenvectors of A. (b) Express the general solution of t' = Az in terms of real valued functions. (c) Sketch the phase portrait of the system. Do not forget to label your axes.
4. Consider the system y'- Ay(t), for t > 0, with A - 1 -2 (a) Show that the matrix A has eigenvalues וג --1and Az--3 with corresponding eigenvectors u (1,1) and u2 (1,-1) (b) Sketch the trajectory of the solution having initial vector y(0) = ul. (c) Sketch the trajectory of the solution having initial vector y(0) -u2. (d) Sketch the trajectory of the solution having initial vector y(0)-u -u 1 U
4. Consider the system y'- Ay(t), for...
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