Consider the system: z'(t) + tr(t) + (t-1 )y(t) = 0, s(t) + (t-1)x(t) + ty(t) = 0, x(0)--4 y(0) = 2 Determine the solution functions, ()y) using ONLY the Fundamental Matrix method. Compute th...
Using the Runge-Kutta fourth-order method, obtain a solution to dx/dt=f(t,x,y)=xy^3+t^2; dy/dt=g(t,x,y)=ty+x^3 for t= 0 to t= 1 second. The initial conditions are given as x(0)=0, y(0) =1. Use a time increment of 0.2 seconds. Do hand calculations for t = 0.2 sec only.
2. Consider the differential equation ty" – (t+1)y' +y = 2t2 t>0. (a) Check that yı = et and y2 = t+1 are a fundamental set of solutions to the associated homogeneous equation. (b) Find a particular solution using variation of parameters.
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...
4. Suppose the matrix equation Az(t) =#(ty has the property that /2 0 0 D =0 1 0 (0 0 -7, and a change of basis matrix given by T 1 1 P = 1 e 1 Compute the solution f(t), and write down the n-th order differential equation associated to the matrix A 4. Suppose the matrix equation Az(t) =#(ty has the property that /2 0 0 D =0 1 0 (0 0 -7, and a change of basis...
x(t)-A()x(t) + B(t)u(t) 2. Consider the following system y(t)-C(t)x(t) where A(t)- 0 1) Derive the state transition matrix Φ(t, τ). 2) Derive the impulse response function g(t, z). x(t)-A()x(t) + B(t)u(t) 2. Consider the following system y(t)-C(t)x(t) where A(t)- 0 1) Derive the state transition matrix Φ(t, τ). 2) Derive the impulse response function g(t, z).
Consider the following CT periodic signals x(t), y(t) and z(t) a(t) 5 -4 y(t) 5/-4 z(t) 5 4 (a) [2 marks] Find the Fourier series coefficients, ak, for the CT signal r(t), which is a periodic rectangular wave. You must use the fundamental frequency of r(t) in constructing the Fourier series representation (b) [2 marks] Find the Fourier series coefficients, bk, for the CT signal y(t) cos(t) You must use the fundamental frequency of y(t) in constructing the Fourier series...
Problem 5 Consider the linear system [1 2 0 2 -4 7x(t) 1 -4 6 y(t) [1 -2 2] (t). (4) a(t = (a) Is the system (4) observable? (b) Give a basis for the unobservable subspace of the system (4). In the remainder of this problem, consider the linear system а — 3 8— 2а 0 1 2a u(t) (t) (5) x(t) = with a a real parameter. (c) Determine all values of a for which the system (5)...
4. Solve the following system of linear equations using the inverse matrix method. 1 y = 1 2 , 3 2 -r- 1 5 4 a) x+y +z= 6 x-y-3z=-8 x+y- 2z=-6 b) Solve the following system of linear equations using Cramer's Rule. 5. 2 1 -X- 3 2 1 3 X+-y-1 5 4 y = 1 a) x+y+z= 6 x-y-3z=-8 x+y- 2z = -6 b) 4. Solve the following system of linear equations using the inverse matrix method. 1...
Solve the following system of linear equations by using the inverse matrix method X+Y+Z=4 -2X-Y+3Z=1 Y+5Z=9
1. Solve the system of equations using Laplace Transform(LT): With IV: x(0) 4 With IV :y (0)-5 a. Apply Laplace transform (LT) to the system and solve, by using elimination method, for x(s), and y(s). b. Apply inverse-Laplace transform (L:'T) to the system of s-functions, then solve for x(t), and y(t) 1. Solve the system of equations using Laplace Transform(LT): With IV: x(0) 4 With IV :y (0)-5 a. Apply Laplace transform (LT) to the system and solve, by using...