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Consider the following nonhomogeneous system for 2-dimensional vector X(t), t 2 0. 0 x(0)1 2 -1 where A- 5 -2 (a) Use 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> and the initial condition the solution u(x, t) by complet...
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>...
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the...
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the initial condition Lee) Find the solution u(z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution ue(x). (b) Denote v(x, t) u(a, t) - e(). Derive the IBVP for the function v(x,t). (c) Find v(x, t) (d) Find u(, t)...
Consider the 3-dimensional system of linear equations 1 1 1 X' = X 2 1 -1...
Consider the 3-dimensional system of linear equations 1 1 1 X' = X 2 1 -1 0-1 1 (a) Find a fundamental set of solutions for this system. Note that -1 is one of the eigenvalues (b) Find the general solution, and use it to find the solution satisfying -4 X(0) 2
Consider the 3-dimensional system of linear equations 1 1 1 X' = X 2 1 -1 0-1 1 (a) Find a fundamental set of solutions for this system....
could someone explain this with helpful workspace? Problem 3. (1 point) Use the Laplace transform to...
could someone explain this with helpful workspace?
Problem 3. (1 point) Use the Laplace transform to solve the following initial value problem: y" +9y' = 0 y(0) = 3, y(0) = 5 a. Using Y for the Laplace transform of y(t), i.e., Y = L{y(t)}, find the equation you get by taking the Laplace transform of the differential equation 0 b. Now solve for Y(S) = c. Write the above answer in its partial fraction decomposition, Y(s) = sta +...
Consider the following IVP y″ + 5y′ + y = f (t), y(0) = 3, y′(0) ...
Consider the following IVP
y″ + 5y′ +
y = f (t), y(0) = 3,
y′(0) = 0,
where
f (t) =
{
8
0 ≤ t ≤ 2π
cos(7t)
t > 2π
(a)
Find the Laplace transform F(s) =
ℒ { f (t)} of f (t).
(b)
Find the Laplace transform Y(s) =
ℒ {y(t)} of the solution y(t)
of the above IVP.
Consider the following IVP y" + 5y' + y = f(t), y(0) = 3, y'(0) =...
(2-2) Problem 1, 12 points Let A and consider the system of differential 1 -2 equations...
(2-2) Problem 1, 12 points Let A and consider the system of differential 1 -2 equations y' = Ay. kx a) Find the characteristic polynomial of A b) Use the Laplace transform method to compute eAt. c) Solve the system y - Ay, 0)o
Verify that the vector X is a particular solution of the given nonhomogeneous linear system x'...
Verify that the vector X is a particular solution of the given nonhomogeneous linear system x' 2 1 Xo - 13 For X, , one has Since the above expressions -Select-- te is a particular solution of the given system
(6). The quantities x(t) and y(t) satisfy the simultaneous equations dt dt dx dt where x(0)-y(0)-...
(6). The quantities x(t) and y(t) satisfy the simultaneous equations dt dt dx dt where x(0)-y(0)-ay (0)-0, and ax (0)-λ. Here n, μ, and λ are all positive real numbers. This problem involves Laplace transforms, has three parts, and is continued on the next page. You must use Laplace transforms where instructed to receive credit for your solution (a). Define the Laplace Transforms X(s) -|e"x(t)dt and Y(s) -e-"y(t)dt Laplace Transform the differential equations for x(t) and y(t) above, and incorporate...
Use the variation of parameters formula to find a general solution of the system x'(0) AX(t)...
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).
1. Solve the system of equations using Laplace Transform(LT): With IV: x(0) 4 With IV :y (0)-5 a....
1. Solve the system of equations using Laplace Transform(LT): With IV: x(0) 4 With IV :y (0)-5 a. Apply Laplace transform (LT) to the system and solve, by using elimination method, for x(s), and y(s). b. Apply inverse-Laplace transform (L:'T) to the system of s-functions, then solve for x(t), and y(t)
1. Solve the system of equations using Laplace Transform(LT): With IV: x(0) 4 With IV :y (0)-5 a. Apply Laplace transform (LT) to the system and solve, by using...
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>...
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the initial condition Lee) Find the solution u(z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution ue(x). (b) Denote v(x, t) u(a, t) - e(). Derive the IBVP for the function v(x,t). (c) Find v(x, t) (d) Find u(, t)...
Consider the 3-dimensional system of linear equations 1 1 1 X' = X 2 1 -1 0-1 1 (a) Find a fundamental set of solutions for this system. Note that -1 is one of the eigenvalues (b) Find the general solution, and use it to find the solution satisfying -4 X(0) 2
Consider the 3-dimensional system of linear equations 1 1 1 X' = X 2 1 -1 0-1 1 (a) Find a fundamental set of solutions for this system....
could someone explain this with helpful workspace?
Problem 3. (1 point) Use the Laplace transform to solve the following initial value problem: y" +9y' = 0 y(0) = 3, y(0) = 5 a. Using Y for the Laplace transform of y(t), i.e., Y = L{y(t)}, find the equation you get by taking the Laplace transform of the differential equation 0 b. Now solve for Y(S) = c. Write the above answer in its partial fraction decomposition, Y(s) = sta +...
Consider the following IVP
y″ + 5y′ +
y = f (t), y(0) = 3,
y′(0) = 0,
where
f (t) =
{
8
0 ≤ t ≤ 2π
cos(7t)
t > 2π
(a)
Find the Laplace transform F(s) =
ℒ { f (t)} of f (t).
(b)
Find the Laplace transform Y(s) =
ℒ {y(t)} of the solution y(t)
of the above IVP.
Consider the following IVP y" + 5y' + y = f(t), y(0) = 3, y'(0) =...
(2-2) Problem 1, 12 points Let A and consider the system of differential 1 -2 equations y' = Ay. kx a) Find the characteristic polynomial of A b) Use the Laplace transform method to compute eAt. c) Solve the system y - Ay, 0)o
Verify that the vector X is a particular solution of the given nonhomogeneous linear system x' 2 1 Xo - 13 For X, , one has Since the above expressions -Select-- te is a particular solution of the given system
(6). The quantities x(t) and y(t) satisfy the simultaneous equations dt dt dx dt where x(0)-y(0)-ay (0)-0, and ax (0)-λ. Here n, μ, and λ are all positive real numbers. This problem involves Laplace transforms, has three parts, and is continued on the next page. You must use Laplace transforms where instructed to receive credit for your solution (a). Define the Laplace Transforms X(s) -|e"x(t)dt and Y(s) -e-"y(t)dt Laplace Transform the differential equations for x(t) and y(t) above, and incorporate...
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).
1. Solve the system of equations using Laplace Transform(LT): With IV: x(0) 4 With IV :y (0)-5 a. Apply Laplace transform (LT) to the system and solve, by using elimination method, for x(s), and y(s). b. Apply inverse-Laplace transform (L:'T) to the system of s-functions, then solve for x(t), and y(t)
1. Solve the system of equations using Laplace Transform(LT): With IV: x(0) 4 With IV :y (0)-5 a. Apply Laplace transform (LT) to the system and solve, by using...
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