4. Consider the system y'- Ay(t), for t > 0, with A - 1 -2 (a) Show that the matrix A has eigenva...
Solving 1. Let -Ar, A constant, with real and distinct eigenvalues 3-1 0 2-2 3 A 2 0 0 (a) Find the eigenvalues and the corresponding eigenvectors for the matrix A. (b) Use (a) to write down a fundamental matrix Φ(t) for the system z' = Az, and use Φ(t) to calculate the solution of this system that satisfies the initial condition (0)0
(1 point) -1 -4 a. Given that V1 [ 2] and U2 --10 are eigenvectors of the matrix _2] determine the corresponding eigenvalues. 4 11 = 12 = = -4x b. Find the solution to the linear system of differential equations x' y' satisfying the initial conditions x(0) = -3 and y(0) = 4. 4x – 2y x(t) = y(t) =
*Note: Please answer all parts, and explain all workings. Thank you! 3. Consider the follo 2 lu The boundary conditions are: u(0,y, t) - u(x, 0,t) - 0, ou (a, y, t) = (x, b, t) = 0 ay The initial conditions are: at t-0,11-4 (x,y)--Yo(x,y) . ot a) Assume u(x,y,t) - X(x)Y(y)T(t), derive the eigenvalue problems: a) Apply the boundary conditions and derive all the possible eigenvalues for λι, λ2 and corresponding eigen-functions, Xm,Yn b) for any combination of...
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?
(1 point) Consider the linear system 3 2 ' = y. -5 -3 a. Find the eigenvalues and eigenvectors for the coefficient matrix. 2 = 15 and 2 V2 b. Find the real-valued solution to the initial value problem Syi ly 3y1 + 2y2, -541 – 3y2, yı(0) = 0, y2(0) = -5. Use t as the independent variable in your answers. yı(t) y2(t)
Slove 2nd problem plz (1) Find the eigenvalues and corresponding eigenvectors of [o1 0 0 0 1 2 1 -2 HINT: Note that 13 + 2/2 - 1 - 2 can be regrouped as 1(12 - 1)+2(12-1). Then factor out the common (12 - 1). (2) Solve the equation Y" + 2y' - - 2y = 0) using the method of converting to a linear system of first-order ODE's. Show that the coefficient matrix is the 3 x 3 matrix...
8. 20 pts.] Suppose that a 2 x2 matrix A has the following eigenvalues and eigenvectors: () 12, 1 r2=1, 2 2 (a) Classify the equilibrium 0 (node, saddle, spiral, center). Is it stable or unstable? (b) Sketch the trajectories of the system A , where a the phase plane below. (c) On the next page, sketch the graphs of r1 (t) and 2(t) versus t for the solution that satisfies the initial condition x1(0) = 1, x2(0) = 1...
(1 point) Consider the linear system "(-1: 1) y. a. Find the eigenvalues and eigenvectors for the coefficient matrix. 1 v1 = and 2 V2 b. For each eigenpair in the previous part, form a solution of y' = Ay. Use t as the independent variable in your answers. (t) = and yz(t) c. Does the set of solutions you found form a fundamental set (i.e., linearly independent set) of solutions? Choose
Previous Problem Problem List Next Problem (1 point) Consider the linear system -3 -2 >= -3) y. 5 3 a. Find the eigenvalues and eigenvectors for the coefficient matrix. di = on = and 12 = · U2 b. Find the real-valued solution to the initial value problem { vi = -3yı – 2y2, 5yı + 3y2, yı(0) = 11, y2(0) = -15. y۔ = Use t as the independent variable in your answers. yı(t) = y2(t) =
(1 point) Consider the linear system 3 y y 5 3 a. Find the eigenvalues and eigenvectors for the coefficient matrix. 2 0 and A2 = -1 02 -3- -3+1 b. Find the real-valued solution to the initial value problem Svi C = -3y - 2y2, 591 +372 y.(0) = 6, 32(0) = -15. Use t as the independent variable in your answers. yı() y2(t) = 0