Partial Differential Equations : 8+1 pts) the following Heat problem or (o,)-1, (1-2 0 e steady state solution y(x). e transient solution ω(x,t) using the corresponding homogenous Heat proble ll...
Problem 7.2 The differential equations for a second-order thermal system are y=x2 where u is the control input. (a) Show that the plant is type zero. As a consequence, the steady-state error using proportional control is non-zero. Find the steady-state error as a function of G (b) To achieve zero steady-state error, integral control will be used, by adding the state variable zo with which is appended to the original equations, making the system third-order. For the resulting third-order system,...
T (1 point) Find the solution to the linear system of differential equations 8.x - 2y 12x - 2y satisfying the initial conditions (0) = -5 and y(0) -13 z(t) = y(t) Note: You can earn partial credit on this problem. preview answers Entered Answer Preview
(15 pts) Given the following differential equations with the initial condition y(0) = 1, determine (1) the zero-input response yzi(t), (2) the zero-state response yzs (t) and (3) the total response y(t) for the input x(t) = e-fu(t) by using Laplace Transform. (5 pts) x+6y(t) = x - x() (1) Yzi(t) = (2) yzs(t) = (3) y(t) = (5 pts) (5 pts) 2. (10 pts) Given the following differential equations, find the total response y(t) if y(0) = 1 for...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0 1. Consider the Partial Differential Equation ot u(0,t) =...
please solve number 4 Problem No.1 Solve the following first order differential equations by finding: a- Homogenous solution a. The particular solution b- The total (complete) solution for the corresponding initial conditions. Note: Answer all questions clearly and completely. 1- y' + 10y = 20; y(0) = 0 2- 4y' - 2y = 8; y(0) = 10 3- 10y' = 200; y(0) = -5 4- 2y' + 8y = 6cos(wt); y(0) = 0. Let o = 12 rads/sec.
Problem 3: Insights into Differential Equations a. Consider the differential equation 습 +4 = f(t), where f(t) = e-u, 12 0. Please write the forms of the natural and forced solution for this differential equation. You DO NOT need to solve. (7 points) b. Again consider the differential equation f(t), where f(t) is an input and y(t) is the output (response) of interest. Please write the differential equation in state-space form. (10 points) c. The classical method for solving differential...
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...
Question 1 [Total 20 marks] (a) [5 marks] In a steady-state two-dimensional heat flow problem, the temperature, u, at any point in the domain (t, ) satisfies the differential equation u y(2-y) u= U0F With the given temperature boundary condition as follows: u(x, 0) = 0, u(x, 2) = x(4-x), 0 < x < 4 Calculate the temperature at the interior points a, b, and c using a mesh size h-1. Question 1 [Total 20 marks] (a) [5 marks] In...
2. (10 pts) Given the following differential equations, find the total response y(t) if y(0) = 1 for the input x(t) = 24 cos(6t) u(t) by using Laplace Transform. how t6yce) = . _ x06) y(t) =
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...