Problem 3 (8 points) (a)Find the natural response and the COMPLEX forced response (2 points). (b) And then write the general REAL solution of the given differential equation (2 points). (c)Rewrit...
Name PROBLEM 2: (18%) Given the following differential equation (a) Find the forced response y(t) to a unit ramp input of u(t). (9%) (Medium) (b) Find the steady-state response yo) subject to u(t) frequency response formula.) (9%) (Easy) 3cos (0.5t-0.5). (Hint: use the Name PROBLEM 2: (18%) Given the following differential equation (a) Find the forced response y(t) to a unit ramp input of u(t). (9%) (Medium) (b) Find the steady-state response yo) subject to u(t) frequency response formula.) (9%)...
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...
please answer b. and c. Problem 1. Consider the differential equation given by (a) On the axes provided below, sketch a slope field for the given differential equation at the nine points indicated. locales de mor t e wold qolution to the given differential equation with the initial condition (b) Let y = f(x) be the particular solution to the given differential equation with the initial condition f(0) = 3. Use Euler's method starting at x = 0, with a...
Problem 3. For the following system, (a) compute the eigenvalues, (b) compute the associated eigenvectors, (c) if the eigenvalues are complex, determine if the origin is a spiral sink, a spiral source, or a center; determine the natural period and natural frequency of the oscillations, and determine the direction of the oscillations in the phase plane, (d) sketch the phase portrait for the system; and (e) compute the general solution. ar dY (1 -3 dt Y, Problem 3. For the...
Question 2: Differential Equations a) (3 points) Find the general solution to the equation. Use C,C1,C2 ... to denote arbitrary constants as necessary. y"(t) = sin6t + 20e b) (5 points) Solve the following separable differential equation for the given initial condition. y')= (1) = 0 c) (5 points) Solve the following first-order linear differential equation for the given initial condition. y't) + 7y - 3,y(0) - 1 d) (2 points) State the equilibrium solution and whether it is stable...
3. Homogeneous linear systems with complex and repeated eigenvalues. Find the general solu- tion of the given system of differential equations. For the two-dimensional systems, classify the origin in terms of stability and sketch the phase plane (a) x'(t) y'(t) 6х — у, 5х + 2y. = (b) 4 -5 x'(i) х. -4 (c) 1 -1 2 x'() -1 1 0x -1 0 1 3. Homogeneous linear systems with complex and repeated eigenvalues. Find the general solu- tion of the...
Problem 1 (14 points) (a) Find the general solution to a third-order linear homogeneous differential equation for y(1) with real numbers as coefficients if two linearly independent solutions are known to be e-21 and sin(3.c). e (b) Determine that differential equation described in part (a).
Problem 1 (14 points) (a) Find the general solution to a third-order linear homogeneous differential equation for y(1) with real numbers as coefficients if two linearly independent solutions are known to be e-21 and sin(3.c). e (b) Determine that differential equation described in part (a).
6. [0/2 points) DETAILS PREVIOUS ANSWERS Find the general (real) solution of the differential equation: y"- 2y'- 15y=-51 sin(3 x) -3x | Ae 5x + Be 34 y(x) = 8.5 + -cos(3x) * 17 51 14 sin(3x) - - Find the unique solution that satisfies the initial conditions: Y(0) = 2.5 and y'(o)=37 y(x) = 7. [-12 Points) DETAILS Find the general (real) solution of the differential equation: y" + 4y' + 4y=64 cos(2x) y(x) = Find the unique solution...
(1 point) a. Find the most general real-valued solution to the linear system of differential equations x -8 -10 x. xi(t) = C1 + C2 x2(t) b. In the phase plane, this system is best described as a source / unstable node sink / stable node saddle center point / ellipses spiral source spiral sink none of these ОООООО (1 point) Calculate the eigenvalues of this matrix: [Note-- you'll probably want to use a calculator or computer to estimate the...