Please do the problem step by step. Thank you very much!
For the following linear systems with given eigenvalues, (a) Determine the type of equilibrium po...
3. Homogeneous linear systems with complex and repeated eigenvalues. Find the general solu- tion of the given system of differential equations. For the two-dimensional systems, classify the origin in terms of stability and sketch the phase plane (a) x'(t) y'(t) 6х — у, 5х + 2y. = (b) 4 -5 x'(i) х. -4 (c) 1 -1 2 x'() -1 1 0x -1 0 1 3. Homogeneous linear systems with complex and repeated eigenvalues. Find the general solu- tion of the...
1. For each of the following systems, (i) determine all critical points, (ii) determine the corresponding linear system near each critical point, and (ii) determine the eigenvalues of each linear system and the corresponding conclusion that can be inferred about the nonlinear system. (a) dz/dt x- - zy, dy/dt 3y- xy-2y (b) dr/dt r2 + y, dy/dt=y-ay
2. Consider the linear system: - (1 2) Y.with initial conditions) Y dt a) Compute the eigenvalues and eigenvectors for the system. b) For each eigenvalue, pick an associated eigenvector V and determine a vector solution y(t) to the system. c) Draw an accurate phase portrait for this system. What type of equilibrium point is the origin?
Consider the nonlinear System of differential equations di dt dt (a) Determine all critical points of the system (b) For each critical point with nonnegative x value (20) i. Determine the linearised system and discuss whether it can be used to approximate the ii. For each critical point where the approximation is valid, determine the general solution of iii. Sketch by hand the phase portrait of each linearised system where the approximation behaviour of the non-linear system the linearised system...
3) Given the systemxx2-x,y'-2y, find all fixed points. For each fixed point linearize the system near the fixed point, shift the fixed point to the origin, determine the eigenvalues of the linearized system, and determine whether the fixed point is a source, sink, saddle, stable orbit, or spiral. Attach a phase plane diagram to verify the behavior you found. 3) Given the systemxx2-x,y'-2y, find all fixed points. For each fixed point linearize the system near the fixed point, shift the...
1. Consider the differential equation" = y2 - 4y - 5. a) Find any equilibrium solution(s). b) Create an appropriate table of values and then sketch (using the grid provided) a direction field for this differential equation on OSIS 3. Be sure to label values on your axes. c) Using the direction field, describe in detail the behavior of y ast approaches infinity. 2. Short answer: State whether or not the differential equation is linear. If it is linear, state...
#20 please and specifically c.) .... but with the initial conditions only being A= (1,-1) and D=(-1,2). For A, I got x(t)=e^(-4t) and y(t) = -e^(-4t). For D, I got x(t)= 3/4*e^(4t)-7/4*e^(-4t) and y(t)=1/4*e^(4t)+7/4*e^(-4t) 295 3.3 Phase Portraits for Linear Systems with Real Eigenvalues 20. The slope field for the system y 3 dx =2x +6y dt dy = 2x - 2y dt is shown to the right. (a) Determine the type of the equilibrium point at the origin. x...
1. The populations of two competing species x(t) and y(t) are governed by the non-linear system of differential equations dx dt 10x – x2 – 2xy, dy dt 5Y – 3y2 + xy. (a) Determine all of the critical points for the population model. (b) Determine the linearised system for each critical point in part (a) and discuss whether it can be used to approximate the behaviour of the non-linear system. (c) For the critical point at the origin: (i)...
1. Classify the stability type of the equilibrium point (0,0) of the following linear planar homogeneous systems (same notation as in the previous homework). -(G3) 1 -3 2 -1 A-(25 4 -5
O SYSTEMS OF EQUATIONS AND MATR.. Gauss-Jordan elimination with ... Consider the following system of linear equations. 5x + 20y=-10 - 6x-28y - 12 Solve the system by completing the steps below to produce echelon form. R, and R, denote the first and second rows, re arrow notation (-) means the expression/matrix on the left expression/matrix on the right once the row operations are TOD:07 (a) Enter the augmented matrix. X (b) For each step below, enter the coefficient for...