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4. Suppose the matrix equation Az(t) =#(ty has the property that /2 0 0 D =0...
3. Consider the differential equation ty" - (t+1)y + y = t?e?', t>0. (a) Find a...
3. Consider the differential equation ty" - (t+1)y + y = t?e?', t>0. (a) Find a value ofr for which y = et is a solution to the corresponding homogeneous differential equation. (b) Use Reduction of Order to find a second, linearly independent, solution to the correspond- ing homogeneous differential equation. (c) Use Variation of Parameters to find a particular solution to the nonhomogeneous differ- ential equation and then give the general solution to the differential equation.
Suppose that А is an mxn matrix with independent columns and the equation Az = 7...
Suppose that А is an mxn matrix with independent columns and the equation Az = 7 is inconsistent. Then the following statements are true. A. The least squares solution to AT = 5 is given by î = (A” A) "A" 7 B. We can reduce the least squares solution Î = (A" A) "A" 5 as follows. î = (A" A) "A" 5 = = A** (AT) 'A 5 = This calculation follows since when matrices This calculation follows...
Problem 4 (Analytical and Computational-20 points) Given a second-order ordinary differential equation: d2f(t) df(t) with the following initial conditions: (O) 1 and ait 0 (Analytical-10 points)...
Problem 4 (Analytical and Computational-20 points) Given a second-order ordinary differential equation: d2f(t) df(t) with the following initial conditions: (O) 1 and ait 0 (Analytical-10 points) Express Equation (1) in state-space form. Cleary write down the A, B, C, and D matrices. Then find the state transition matrix and determine the solution for f(t) if the input function r(t) is a unit step function. a) b) (Computational-10 points) Write a MATLAB-Simulink program to find the computational solution for f(t) in...
2. Consider the differential equation ty" – (t+1)y' +y = 2t2 t>0. (a) Check that yı...
2. Consider the differential equation ty" – (t+1)y' +y = 2t2 t>0. (a) Check that yı = et and y2 = t+1 are a fundamental set of solutions to the associated homogeneous equation. (b) Find a particular solution using variation of parameters.
3. Consider the differential equation ty" - (t+1)yy = te2, t> 0. ert is a solution...
3. Consider the differential equation ty" - (t+1)yy = te2, t> 0. ert is a solution to the corresponding homogeneous (a) Find a value of r for which y = differential equation (b) Use Reduction of Order to find a second, linearly independent, solution to the correspond- ing homogeneous differential equation
7. (10 points) Find a particular solution yp(t) to the nonhomogeneous equation ty + y -...
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
2. The angular displacement e(t) of a damped forced pendulum of length 1 swinging in a...
2. The angular displacement e(t) of a damped forced pendulum of length 1 swinging in a vertical plane under the influence of gravity can be modelled with the second order non-homogeneous ODE 0"(t) + 270'(t) +w20(t) = f(t), (2) where wa = g/l. The second term in the equation represents the damping force (e.g. air resistance) for the given constant 7 > 0. The model can be used to approximate the motion of a magnetic pendulum bob being driven by...
Question 5 Suppose that A is the matrix (51 -1 A= 0 4 0 1 1...
Question 5 Suppose that A is the matrix (51 -1 A= 0 4 0 1 1 3 (a) Find an invertible matrix P such that P-1AP = J where J is a Jordan normal form. Now consider the system of differential equations f'(t) = 5f(t) +g(t) - h(t), 9'(t) = 49(t) and h'(t) = f(t)+g(t) + 3h(t). (b) Find the general solution to this system of differential equations. (c) What is the dominant long-term behaviour of this system? For what...
Given the system of differential equations o y (7tcos(tut) Write the first order matrix different...
Given the system of differential equations o y (7tcos(tut) Write the first order matrix differential equation that is the basis for using Euler's method to compute the numerical solution. It is assumed you will use two auxiliary functions, xi and t2 Define the functions i and 2 in terms of v and y. E2 dri (t) dt 1(t) dr2(t) dt a2(t)
Given the system of differential equations o y (7tcos(tut) Write the first order matrix differential equation that is the...
Consider the system: z'(t) + tr(t) + (t-1 )y(t) = 0, s(t) + (t-1)x(t) + ty(t) = 0, x(0)--4 y(0) = 2 Determine the solution functions, ()y) using ONLY the Fundamental Matrix method. Compute th...
Consider the system: z'(t) + tr(t) + (t-1 )y(t) = 0, s(t) + (t-1)x(t) + ty(t) = 0, x(0)--4 y(0) = 2 Determine the solution functions, ()y) using ONLY the Fundamental Matrix method. Compute the values (1), y(2)
Consider the system: z'(t) + tr(t) + (t-1 )y(t) = 0, s(t) + (t-1)x(t) + ty(t) = 0, x(0)--4 y(0) = 2 Determine the solution functions, ()y) using ONLY the Fundamental Matrix method. Compute the values (1), y(2)
3. Consider the differential equation ty" - (t+1)y + y = t?e?', t>0. (a) Find a value ofr for which y = et is a solution to the corresponding homogeneous differential equation. (b) Use Reduction of Order to find a second, linearly independent, solution to the correspond- ing homogeneous differential equation. (c) Use Variation of Parameters to find a particular solution to the nonhomogeneous differ- ential equation and then give the general solution to the differential equation.
Suppose that А is an mxn matrix with independent columns and the equation Az = 7 is inconsistent. Then the following statements are true. A. The least squares solution to AT = 5 is given by î = (A” A) "A" 7 B. We can reduce the least squares solution Î = (A" A) "A" 5 as follows. î = (A" A) "A" 5 = = A** (AT) 'A 5 = This calculation follows since when matrices This calculation follows...
Problem 4 (Analytical and Computational-20 points) Given a second-order ordinary differential equation: d2f(t) df(t) with the following initial conditions: (O) 1 and ait 0 (Analytical-10 points) Express Equation (1) in state-space form. Cleary write down the A, B, C, and D matrices. Then find the state transition matrix and determine the solution for f(t) if the input function r(t) is a unit step function. a) b) (Computational-10 points) Write a MATLAB-Simulink program to find the computational solution for f(t) in...
2. Consider the differential equation ty" – (t+1)y' +y = 2t2 t>0. (a) Check that yı = et and y2 = t+1 are a fundamental set of solutions to the associated homogeneous equation. (b) Find a particular solution using variation of parameters.
3. Consider the differential equation ty" - (t+1)yy = te2, t> 0. ert is a solution to the corresponding homogeneous (a) Find a value of r for which y = differential equation (b) Use Reduction of Order to find a second, linearly independent, solution to the correspond- ing homogeneous differential equation
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
2. The angular displacement e(t) of a damped forced pendulum of length 1 swinging in a vertical plane under the influence of gravity can be modelled with the second order non-homogeneous ODE 0"(t) + 270'(t) +w20(t) = f(t), (2) where wa = g/l. The second term in the equation represents the damping force (e.g. air resistance) for the given constant 7 > 0. The model can be used to approximate the motion of a magnetic pendulum bob being driven by...
Question 5 Suppose that A is the matrix (51 -1 A= 0 4 0 1 1 3 (a) Find an invertible matrix P such that P-1AP = J where J is a Jordan normal form. Now consider the system of differential equations f'(t) = 5f(t) +g(t) - h(t), 9'(t) = 49(t) and h'(t) = f(t)+g(t) + 3h(t). (b) Find the general solution to this system of differential equations. (c) What is the dominant long-term behaviour of this system? For what...
Given the system of differential equations o y (7tcos(tut) Write the first order matrix differential equation that is the basis for using Euler's method to compute the numerical solution. It is assumed you will use two auxiliary functions, xi and t2 Define the functions i and 2 in terms of v and y. E2 dri (t) dt 1(t) dr2(t) dt a2(t)
Given the system of differential equations o y (7tcos(tut) Write the first order matrix differential equation that is the...
Consider the system: z'(t) + tr(t) + (t-1 )y(t) = 0, s(t) + (t-1)x(t) + ty(t) = 0, x(0)--4 y(0) = 2 Determine the solution functions, ()y) using ONLY the Fundamental Matrix method. Compute the values (1), y(2)
Consider the system: z'(t) + tr(t) + (t-1 )y(t) = 0, s(t) + (t-1)x(t) + ty(t) = 0, x(0)--4 y(0) = 2 Determine the solution functions, ()y) using ONLY the Fundamental Matrix method. Compute the values (1), y(2)
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