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15. Show that the stability of any solution x(1) of the nonhomogeneous equation * - Ax...
Use the variation of parameters formula to find a general solution of the system x'(0) AX(t) + f(t), where A and f(t) are given -4 2 А. FU) 21 12 +21 Let x(t) = xy()+ X(t), where x, (t) is the general solution corresponding to the homogeneous system, and X(t) is a particular solution to the nonhomogeneous system. Find X. (t) and X.(1).
Consider the following 2nd order nonhomogeneous linear equation
x 00 + 4x 0 + 5x = cos 2t
1. Solve for the fundamental solutions of its associated
homogeneous equation.
2. Find a particular solution of the nonhomogeneous
equation.
3. Based on your answer to the previous two questions, write
down the general solution of the nonhomogeneous equation.
Problem II (15 points) Consider the following 2nd order nonhomogeneous linear equation x" + 40' + 5x = cos 2t 1. (6 points)...
Problem 1. Consider the nonhomogeneous heat equation for u(,) subject to the nonhomogeneous boundary conditions 14(0,t) 1, u(r,t)-0,t> and the initial condition the solution u(x, t) by completing each of the following steps (a) Find the equilibrium temperature distribution u ( (b) Denote v, t)t) - u(). Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t)
Problem 1. Consider the nonhomogeneous heat equation for u(,) subject to the nonhomogeneous boundary conditions 14(0,t) 1, u(r,t)-0,t>...
Consider the following nonhomogeneous linear differential equation ay 6) + by(s) + cy!4) + dy'"' + ky'' + my' + ny=3x²3x - 7cos +1 where coefficients a, b, c, d, k, m, n are constant. Assume that the general solution of the associated homogeneous linear differential equation is YAEC,+Ce**+ c xe** + c.xe3* + ecos What is the correct form of the particular solution y of given nonhomogeneous linear differential equation? Yanitiniz: o Yo=Ax*e** + Ex + F **+Cxcos() +oxsin()+Ex+F...
Problem 1. Consider the nonhomogeneous heat equation for u(x,t) subject to the nonhomogeneous boundary conditions and the initial condition e solution u(z, t) by completing each of the following steps Find the equilibrium temperature distribution ue(a) (b) Denote v(, t)t) -)Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x,t)
Problem 1. Consider the nonhomogeneous heat equation for u(x,t) subject to the nonhomogeneous boundary conditions and the initial condition e solution u(z, t) by completing each...
Engineering Mathematics 1 Page 3 of 10 2. Consider the nonhomogeneous ordinary differential equation ry" 2(r (x - 2)y 1, (2) r> 0. (a) Use the substitution y(x) = u(x)/x to show that the associated homogeneous equation ry" 2(r (x - 2)y 0 transforms into a linear constant-coefficient ODE for u(r) (b) Solve the linear constant-coefficient ODE obtained in Part (a) for u(x). Hence show that yeand y2= are solutions of the associated homogeneous ODE of equation (2). (c) Use...
7. (10 points) Find a particular solution yp(t) to the nonhomogeneous equation ty + y - y = 24t*, t> 0, given the fact that the general solution of the associated homogeneous equation is yn(t) = cit + cat-, C1, C2 E R
Part A is first 2 lines, Part B is last 2 lines, thanks!
For Problems 13-17, find a particular solution of the nonhomogeneous equation, given that the functions y(x) and y2(x) are linearly independent solutions ofthe corresponding homogeneous equation. Note: The cocfficient of y" must always be 1, and hence a preliminary division may be required y2(x) = x-2 ·y1(x) = x y2(x) = ex
For Problems 13-17, find a particular solution of the nonhomogeneous equation, given that the functions...
2. Given the nonhomogeneous 2nd order differential equation y" +2y = xe*: (8 pts) a. Identify the forcing function (ie. the nonhomogeneous term we call f(x)). b. Write the homogeneous equation associated with this DE. c. Find the particular solution to the homogeneous DE from part b which satisfies the initial conditions y(0) = 2, y'(O)=-1. (note: you will NOT be using technique of undetermined coefficients)
Find the solution by Fourier series of the heat equation with nonhomogeneous boundary conditions. Assume that the initial condition is given byf(x) uo u(0,t) = uo, u(L,t) = u1, t > 0 u(x,0) = f(x), 0 < x < L