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Question 7: Solve the entire problem using Laplace Transforms. Recall the DE for our two-vessel water clock ах - Ax, where A dt k(0)= DE IC: -1] Let X(s) denote the Laplace transform of x(t). Then x(s) = (sl-A)-1 (0) There is no forcing term, so this is just the zero-input or homogeneous solution. Solve for X(s) and record your answer in the answer template. The first component has been given for you Question 7: The solution in the transform domain is: and X2(s) k(s) = [k(s) where X,() S+1 Question 8: Integrating the Flows We saw earlier, that the situation is clearer if we plot the cumulative outflows fi(t) and f2 (t), where h(t) = ft yı (t) dt f2(t) = y2(t) dt and Now recall identity #32 in our Table of Laplace transforms for the transform of an integral Integration in the time domain is simply division by s in the transform domain. F(s) F (s) If the transform of f(t) is F(s), then the transform of its integral (from 0 tois Just divide by s!! fi(t) , in two steps f2 (t) Find the cumulative outflow vector f(t) = fi(t) 2 (t) First define F(s) then apply MATLAB's command ilaplace to find f(t Complete this code to find the f(t) vector in the time domain. The answer should be the same as above Question 8: Complete this code to find the cumulative flow vector f(t) using Laplace transforms syms s A--1 0; 1-1] % system matrix % initial conditions % find X here using inv () % find F here. Integration is division by s.

Answer 1

(7) Given: A = (s) = (sl-A)-T(0) Using the above, we have X(s)-S 01-11 -1 S+1 01 L111 Using: A- (s+1)L1 s+1]1 s+1 s +1 s +1 Xi (s) X2(s) Therefore, X(s)-1281 whereč(s)-s+1 s +2 and X2 (s)- (s+