III. Consider the following state equations: 0 -1 x=11-2 with 2(0) 11 10]T and a(t) t, t-0. Solve...
Consider the following parametric equations. x = √1 + 2 , y = 2√t; 0 ≤ t ≤ 16 a. Eliminate the parameter to obtain an equation in x and y. b. Describe the curve and indicate the positive orientation.
Consider the system of linear differential equations z,(t)-17/11 z(t) + 9/11 y(t) y,(t)-18/11 z(t) + 38/11 y(t) a) Find the equation of the x-nullcline. Write your answer as an equation in z and y Answer b) Find the equation of the y-nullcline. Write your answer as an equation of z and y Answer. c) The nullclines divide the plane into four regions as illustrated below. 忽聡 2 -2 2 -2 For each of the regions, determine the direction of the...
7. Solve the following differential equations. dy 2 y= 5x, x>0. + a) dx dx 1+2x 4e', t>0 b) t dt
7. Solve the following differential equations. dy 2 y= 5x, x>0. + a) dx dx 1+2x 4e', t>0 b) t dt
Discretization, ODE solving, condition number. Consider the differential equation 5y"(x) - 2y'(x) +10y(x)0 on the interval x E [0,10] with boundary conditions y(0)2 and y (10) 3 we set up a finite difference scheme as follows. Divide [0,10] into N-10 sub-intervals, i.e. {xo, X1, [0,1,. 10. Denote xi Xo + ih (here, h- 1) and yi E y(x). Approximate the derivatives as follows X10- 2h we have the following equations representing the ODE at each point Xi ,i = 1,...
1 & 5
Solve the following heat equations using Fourier series ux Ut, 0 <x <1,t>0, u (0,t) = 0 = u(1,t), u(x,0) = x/2 1/ 2/ Ux=Ut, 0<x< m ,t>0 ,u(0,t) = 0 = u( 1, t), u(x, O) = sinx- sin3x 3/ usxut, O <x < 1 ,t>0, u(0,t) = 0 = u,(1, t), u(x,0) = 1 -x2 Ux=Ut,O<x <m ,t>0, u(0, t) = 0 = u,( rt , t) , u(x, 0) = (sinxcosx)2 4/ 5/Solve the...
A system is described by the state variable equations 0 -2 1 10 y(t) =[1 0 (ls(t). Determine G(s) = r(s)/U(s).
(10 points) Consider the following system of linear equations. 2x1 + 4x2 - X3 = 0 31 +2302 + x3 = 3 (a) Write the system as a vector equation in which the left-hand-side is a linear combination of column vectors. (b) Find the solution set of the system in vector form. Check that every solution is the sum of a particular solution and a vector in the null space of the coefficient matrix. (c) Find a basis for the...
1-1 11?? (c) Consider the system of linear equations | 3 1 40-1 | x = | 2 | , where a 2 a a+1 is a scalar. (i) 1 (ii) Determine the value(s) of a such that the system is consistent with infinitely many solutions; consistent with one and only one solution; and , (iii) inconsistent. Solve the system when it is consistent. 20 marks
1. Write the state-space equations for the system shown below ri (t) +2 (t) u (t) Figure 1: System of Problem#1 2. Evaluate the state transition matrix eA for the matrix below and find the homogenous solution given x (0) 1 1 ] A=10-21 3. Find the power lution in powers of x. Show the details of your work. s (b) y" +4y=0 4. Determine if either the Frobenus or regular power series could be the method of your choice...
please solve 8-14
8-13. Given the dynamic equations ast) = Ax(t)+ Bu(t) y(t)=Cx(t) I 0 2 0 1 To A = 120 B= 1 C= (a) 1 -1 0 1 [ 0 2 0 1 1 A = 120 B c=1017 (b) (-1 11] -2 1 0 1 A- 7 -2 0 B- C-[1 0 0] A=0 (d) [ 00 -1 832 -} - ic-[1 0] (e) -2 -3 8-14. For the systems described in Prob. 8-13, find the transformation...