find the solution of the inhomogeneous system for y" +p(t)y' +q(t)y = f(t), a second order scalar equation with p, q, f continuous on interval I,
for which (to ) = 0, to on I
find the solution of the inhomogeneous system for y" +p(t)y' +q(t)y = f(t), a second order...
consider the variation of constants formula where P(t)= a) show that solves the initial value problem x'+p(t)=(t) x()= when p and q are continuous functions of t on an interval I and tg p(s)ds We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image tg p(s)ds
Consider the boundary value problem for the general second-order equation with constant coefficients y(a)=YA, y(b)=YB Let the interval a<x<b divided into n subintervals of width h=(b-a)/n.Using central difference approximations find the lineer system that must be solved to approximate y2,y3,,,yn We were unable to transcribe this image01.2 h2 2h We were unable to transcribe this imageProblem 3 boundary value problem for the general second-order equation with constant coefficients dy dy y(a) YA, ybYB. Let the interval a s b be...
An autonomous system of two first order differential equations can be written as: A third order explicit Runge-Kutta scheme for an autonomous system of two first order equations is Consider the following second order differential equation, Use the Runge-Kutta scheme to find an approximate solutions of the second order differential equation, at t = 1.2, if the step size h = 0.1. Maintain at least eight decimal digit accuracy throughout all your calculations. You may express your answer as a...
Q1) Let X(t) be a zero-mean WSS process with X(t) is input to an LTI system with Let Y(t) be the output. a) Find the mean of Y(t) b) Find the PSD of the output SY(f) c) Find RY(0) ------------------------------------------------------------------------------------------------------------------------- Q2) The random process X(t) is called a white Gaussian noise process if X(t) is a stationary Gaussian random process with zero mean, and flat power spectral density, Let X(t) be a white Gaussian noise process that is input to...
Find a second order linear equation L(y) = f(t) with constant coefficients whose general solution is: @ y=Cje24 + C261 + te3t @ (a) The solution contains three parts, so it must come from a nonhomogeneous equation. Using the two terms with undefined constant coefficients, find the characteristic equation for the homogeneous equation. (b) Using the characteristic equation find the homogeneous differential equation. This should be the L(y) we're looking for. (c) Since we have used two terms from the...
Consider the differential equation, L[y] = y'' + p(t)y' + q(t)y = 0, (1) whose coefficients p and q are continuous on some open interval I. Choose some point t0 in I. Let y1 be the solution of equation (1) that also satisfies the initial conditions y(t0) = 1, y'(t0) = 0, and let y2 be the solution of equation (1) that satisfies the initial conditions y(t0) = 0, y'(t0) = 1. Then y1 and y2 form a fundamental set...
A system of two first order differential equations can be written as: A second order explicit Runge-Kutta scheme for the system of two first order equations is Consider the following second order differential equation: Use the Runge-kutta scheme to find an approximate solution of the second order differential equation, at x = 0.2, if the step size h = 0.1. Maintain at least eight decimal digit accuracy throughout all your calculations. You may express your answer as a five decimal...
Q. Determine whether the given functions are exponentially bounded and piecewise continuous on 0 ≤ t < ∞. (a) f(t) = tant (b) f(t) = cosh2t (c) f(t) = , where denotes the greatest integer less than or equal to t. We were unable to transcribe this imageWe were unable to transcribe this image
Question 5 (Unit 6) - 31 marks (a) Express the following inhomogeneous system of first-order differential equations for x(t) and y(t) in matrix form: = 2x + y + 3e", y = 4x – y. Write down, also in matrix form, the corresponding homogeneous system of equations. (b) Find the eigenvalues of the matrix of coefficients and an eigenvector corresponding to each eigenvalue. (c) Hence write down the complementary function for the system of equations. (d) Find a particular integral...
Question 5 (Unit 6) - 31 marks (a) Express the following inhomogeneous system of first-order differential equations for x(t) and y(t) in matrix form: = 2x + y + 3e", y = 4x – y. Write down, also in matrix form, the corresponding homogeneous system of equations. (b) Find the eigenvalues of the matrix of coefficients and an eigenvector corresponding to each eigenvalue. (c) Hence write down the complementary function for the system of equations. (d) Find a particular integral...