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Answer all questions QUESTION FOUR [25 MARKS] (a) Prove that if (t, to) is the transition...
Answer all questions (especially part b if unable to do part a) QUESTION FOUR 25 MARKS]...
Answer all questions (especially part b if unable to do part
a)
QUESTION FOUR 25 MARKS] Consider the initial value problem d x()= A(0)x(t) +f(t), x(0) = xo where xo is some constant vector. A(T)dr A(t)= A(t)A()dr). Show that the matrix X (e) = ei A(0d (a) Assuming satisfies the matrix differential equation: Xt) = A(€)X(1) 10 Maris) (b) Obtain a solution to the initial value problem, given 0 A= f) x(0) -1 3 15 Marks
QUESTION FOUR 25 MARKS]...
Answer All questions please QUESTION FOUR Consider the initial value problem x(t) A(t)x(t) f(t), dt where...
Answer All questions please
QUESTION FOUR Consider the initial value problem x(t) A(t)x(t) f(t), dt where xo is some constant vector (a) Show that the associated homogenous system, x(t) A(t)x(t), has its transi tion matrix as X(t)e Ar)dr provided AeJtr)-Ar) A(t) for all t 10 Marks A(t) A(T)dr (b) Obtain a solution to the initial value problem, given that ()- () 4 A = 6 et and x(0) 1 15 Marks f(t)
QUESTION FOUR Consider the initial value problem x(t)...
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r)) =...
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r)) = K for some constant K is orthogonal to the tangent vector T() of each curve C described by the vector function on the surface passing through Po (xo,yo, zo). Hint, remember that the tangent vector T(o) R'(), so prove that Vfo R'O) 0
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r))...
1 Let f (t), g(t) be a continuous function on some interval I, and to e I. Prove that the initial value problem y'(t) f(t)y + g(t)y2, y(to) zo has a unique and continuous solution φ(t) on a s...
1 Let f (t), g(t) be a continuous function on some interval I, and to e I. Prove that the initial value problem y'(t) f(t)y + g(t)y2, y(to) zo has a unique and continuous solution φ(t) on a small interval containing to, φ(t) satisfies the initial condition φ(to) = to.
1 Let f (t), g(t) be a continuous function on some interval I, and to e I. Prove that the initial value problem y'(t) f(t)y + g(t)y2, y(to) zo has...
() At)x()B(f)u() Consider the following time-varying system y(t) C(f)x(t) where x) R", u(t)E R R 1...
() At)x()B(f)u() Consider the following time-varying system y(t) C(f)x(t) where x) R", u(t)E R R 1 1) Derive the state transition matrix D(t,r) when A(f) = 0 0 sint 2) Assume that x(to) = x0 is given and u(f) is known in the interval [to, 4] Based on these assumptions, derive the complete solution by using the state transition matrix D(f, r). Also show that the solution is unique in the interval [to, 4]. 3) Let x(1) 0 and u(f)...
True or False Ivp questions a) An IVP of the for y' + p(t)y = g(t),...
True or False Ivp questions
a) An IVP of the for y' + p(t)y = g(t), y(0) = yo, with p and g continuous functions defined for all tER, always has a unique differentiable solution y(t) defined for all t E R. b) To find the solution of y' + p(t)y = gi(t) + 92(t), y(0) = yo it suffices to solve y' + p(t)y = gi(t), y(0) = 0 and y' + p(t)y = 92(t), y(0) = 1 and...
Prove that the following two-point boundary-value problem has a UNIQUE solution. Thank you Theorem on Unique Solution, Boundary-Value Problem Let f be a continuous function of (t, s), where 0stSl and...
Prove that the following two-point boundary-value problem has a
UNIQUE solution.
Thank you
Theorem on Unique Solution, Boundary-Value Problem Let f be a continuous function of (t, s), where 0stSl and-00<s< 00. Assume that on this domain THEOREM4 11. Prove that the following two-point boundary-value problem has a unique solution: "(t3 5)x +sin t
Theorem on Unique Solution, Boundary-Value Problem Let f be a continuous function of (t, s), where 0stSl and-00
Question 1 [22 marks] (Chapt ers 2, 3, 4, 5, and 6) Let A e Rn...
Question 1 [22 marks] (Chapt ers 2, 3, 4, 5, and 6) Let A e Rn be an (n x n) matrix and be R. Consider the problem 1 (P2) min2+ s.t. xe R" 1Ax-bil2 1 where & > O is fixed and Il IIl denot es the 2-norm. Call g.(x)=l|2 the objective function of problem (P2) 1Ax-bl2 i) [3 marks] Compute the gradient of g, and use it to show that the solution xi of this problem verifies (I+EATA)(x)...
EXERCISE 3.20 Consider the problem ut =u" + u for u(0,t) u(1, t) 0, u(x,0) f(x). ze(0, 1), t>0, S...
partial differential equations
EXERCISE 3.20 Consider the problem ut =u" + u for u(0,t) u(1, t) 0, u(x,0) f(x). ze(0, 1), t>0, Show that dt and conclude that Use this estimate to bound the difference between two solutions in terms of the difference between the initial functions. Does this problem have a unique solution for each initial function f?
EXERCISE 3.20 Consider the problem ut =u" + u for u(0,t) u(1, t) 0, u(x,0) f(x). ze(0, 1), t>0, Show that...
PHYS1047 a) Given a random variable x, with a continuous probability distribution function fx) 4 marks...
PHYS1047 a) Given a random variable x, with a continuous probability distribution function fx) 4 marks b) The life expectancy (in days) of a mechanical system has a probability density write down equations for the cumulative distribution C(x) and the survival distribution Px). State a relationship between them. function f(x)=1/x, for x21, and f(x)=0 for x <1. i Find the probability that the system lasts between 0 and I day.2 marks i) Find the probability that the system lasts between...
Answer all questions (especially part b if unable to do part
a)
QUESTION FOUR 25 MARKS] Consider the initial value problem d x()= A(0)x(t) +f(t), x(0) = xo where xo is some constant vector. A(T)dr A(t)= A(t)A()dr). Show that the matrix X (e) = ei A(0d (a) Assuming satisfies the matrix differential equation: Xt) = A(€)X(1) 10 Maris) (b) Obtain a solution to the initial value problem, given 0 A= f) x(0) -1 3 15 Marks
QUESTION FOUR 25 MARKS]...
Answer All questions please
QUESTION FOUR Consider the initial value problem x(t) A(t)x(t) f(t), dt where xo is some constant vector (a) Show that the associated homogenous system, x(t) A(t)x(t), has its transi tion matrix as X(t)e Ar)dr provided AeJtr)-Ar) A(t) for all t 10 Marks A(t) A(T)dr (b) Obtain a solution to the initial value problem, given that ()- () 4 A = 6 et and x(0) 1 15 Marks f(t)
QUESTION FOUR Consider the initial value problem x(t)...
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r)) = K for some constant K is orthogonal to the tangent vector T() of each curve C described by the vector function on the surface passing through Po (xo,yo, zo). Hint, remember that the tangent vector T(o) R'(), so prove that Vfo R'O) 0
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r))...
1 Let f (t), g(t) be a continuous function on some interval I, and to e I. Prove that the initial value problem y'(t) f(t)y + g(t)y2, y(to) zo has a unique and continuous solution φ(t) on a small interval containing to, φ(t) satisfies the initial condition φ(to) = to.
1 Let f (t), g(t) be a continuous function on some interval I, and to e I. Prove that the initial value problem y'(t) f(t)y + g(t)y2, y(to) zo has...
() At)x()B(f)u() Consider the following time-varying system y(t) C(f)x(t) where x) R", u(t)E R R 1 1) Derive the state transition matrix D(t,r) when A(f) = 0 0 sint 2) Assume that x(to) = x0 is given and u(f) is known in the interval [to, 4] Based on these assumptions, derive the complete solution by using the state transition matrix D(f, r). Also show that the solution is unique in the interval [to, 4]. 3) Let x(1) 0 and u(f)...
True or False Ivp questions
a) An IVP of the for y' + p(t)y = g(t), y(0) = yo, with p and g continuous functions defined for all tER, always has a unique differentiable solution y(t) defined for all t E R. b) To find the solution of y' + p(t)y = gi(t) + 92(t), y(0) = yo it suffices to solve y' + p(t)y = gi(t), y(0) = 0 and y' + p(t)y = 92(t), y(0) = 1 and...
Prove that the following two-point boundary-value problem has a
UNIQUE solution.
Thank you
Theorem on Unique Solution, Boundary-Value Problem Let f be a continuous function of (t, s), where 0stSl and-00<s< 00. Assume that on this domain THEOREM4 11. Prove that the following two-point boundary-value problem has a unique solution: "(t3 5)x +sin t
Theorem on Unique Solution, Boundary-Value Problem Let f be a continuous function of (t, s), where 0stSl and-00
Question 1 [22 marks] (Chapt ers 2, 3, 4, 5, and 6) Let A e Rn be an (n x n) matrix and be R. Consider the problem 1 (P2) min2+ s.t. xe R" 1Ax-bil2 1 where & > O is fixed and Il IIl denot es the 2-norm. Call g.(x)=l|2 the objective function of problem (P2) 1Ax-bl2 i) [3 marks] Compute the gradient of g, and use it to show that the solution xi of this problem verifies (I+EATA)(x)...
partial differential equations
EXERCISE 3.20 Consider the problem ut =u" + u for u(0,t) u(1, t) 0, u(x,0) f(x). ze(0, 1), t>0, Show that dt and conclude that Use this estimate to bound the difference between two solutions in terms of the difference between the initial functions. Does this problem have a unique solution for each initial function f?
EXERCISE 3.20 Consider the problem ut =u" + u for u(0,t) u(1, t) 0, u(x,0) f(x). ze(0, 1), t>0, Show that...
PHYS1047 a) Given a random variable x, with a continuous probability distribution function fx) 4 marks b) The life expectancy (in days) of a mechanical system has a probability density write down equations for the cumulative distribution C(x) and the survival distribution Px). State a relationship between them. function f(x)=1/x, for x21, and f(x)=0 for x <1. i Find the probability that the system lasts between 0 and I day.2 marks i) Find the probability that the system lasts between...
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