1) Find the fundamental matrix (t) to the system x' (t) = Ax (t) with 1 4-63] A 5 and sketch the phase portrait. =
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![1) Find the fundamental matrix (t) to the system x' (t) = Ax (t) with
1
-1
4-[3]
A
5
and sketch the phase portrait.
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- Match each linear system with one of the phase plane direction fields. (The blue lines are the arrow shafts, and the black dots are the arrow tips.) ? ✓ | 1. z ' = || a' ? 2. ': = ? 3.' = 4. a: = 11 8] -10 3 1 5 -2 1 -5 -13 10] -10 x2 A x2 с x1 (x2 B 2x2/ D Note: To solve this problem, you only need to compute eigenvalues. In fact, it is enough to just compute whether the eigenvalues are real or complex and positive or negative.5) Find dx/dt of Matrix: X 5e-1 2e-¹ -7e-¹Write the given linear system without the use of matrices. Assume X = dx dy dt 6 - (-; 3)× +(-;)** -1 11 X'=( = 6x + 4y + 1 x+3y-1 X (Assume X = (x)) X
- dz (b) Determine the by using the Chain rule with the following information: ds z = cos (rªy^) – sin (z+ 2x°y³) x = y = 8s3 a - The first digit of your matrix number c- The last second digit of your matrix number For example, a student with the matrix number CD200008 will have the values of a = 2 and c = 8. Use 1 if the last digit is of your matrix number is 0. Zero marks will be given without showing the steps for solution.Write a'= -3x+y+ln(t), y'= t'a in matrix form as 188) and f(t) is 181 [*] -= P(1) [*] + F(t) where P(t) isFind two linearly independent solutions of 2a?y" – xy' + (5x + 1)y= 0, x > 0 of the form Y1 = x" (1+ a1r + a2x? + a3x³+..) Y2 = x" (1+ bịT + bzx² + b3x³+..) where r1 > r2- Enter T1 a1 a2 a3 r2 = %3D b2 b3
- Find a (real) fundamental matrix for the system x' = Ax given the following A matrices. Sketch the phase portraits in cach case, including several curves and the direction of (:) ) incrcasing time. Find the (real) solution when x(0) -1 -2 A= 8. -1 (b) -1 -2 A = -8Consider the system of differential equations = (-3/2 3/2)x+ (-¹1) ¹ x' = -3/2 -1 where two eigenvalues of 2 3/2 -3/2 -1 are r = 1/2 and r₂ = 1/2, and the corresponding eigenvector is v= -(2/3). Let x = Ty where T is a transformation matrix. Find two first order differential equations, eigenvector is u=== and = -/+nte, n =+², ₂ -+-+ M and the generalizedFind sin(A), A^5, and expm(A) using (1) Cayley-Hamilton theorem and (2) Jordan decompositiongiven the following matrices: A = [-3 1; 0 -2] A = [2 3; 0 2] Solve by hand, but you can verify it using software afterwards. Optional: plot the phase plane of the above systems xdot=A*x using any software.
- Let B = {(1, 3), (-2,-2)} and B' = {(-12, 0), (-4, 4)} be bases for R², and let 23 = [33] 04 A = R2 relative to B. (a) Find the transition matrix P from B' to B. be the matrix for T: R² ->>> P = 6 9 [V] B [T(V)]B = (b) Use the matrices P and A to find [v] and [T(v)]B, where [V] B¹ = [-4 3]. -12 -1/3 -24 -96 4 -96 4 11 ← (c) Find P-1 and A' (the matrix for T relative to B'). 1/3Consider the continuous-time dynamical system given by the ODE & = -x. (a) Determine the explicit solution and sketch the dynamics in both the phase space and the extended phase space. (b) Determine the equilibria.(b) Now express the system as a first order differential equation x' = A x, where A is given by A = () Find the similarity transformation matrix T and diagonalize A. Find the Fundamental matrix for this system.
