April 2008

April 2008 - (b) Find a basis for the null space (also...

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MATH-263 1. (12 marks) Solve the initial value problem April 2008 1 sin(x) :: + 2ycos(x) = 4 cos(x) , Y(7r/2) == O. On what interVal is this solution valid? 2. (10 marks) Solve implicitly 3. (12 marks) Solve the initial valu~ problem y" - 4y' +5y = 2sinx, 4. (10 marks) Find the general solution of y(O) = 0, y'(O) = o. x>O. (0) 5. (12 marks) Find the general solution of d4 d3 .(}? -.!!. + 2-.!!. + -.!!. = 1 + ge2x. dx4 dx3 dx2 6. (to marks) Using Laplace transforms, and making use .ofthe table below as needed, solve y" + 2y~ t5(t ~ 7r) with initial conditions y(O) = 1,y'(0) = 2. I function f(t) I--L-a-pl-ace-t-rans----:fo-rm-F-(,. ...,s ),------------.1 1 l/s (8)0) tn nl/sn+1 (s > 0) eat 1/(8- a) (8) a) sin at a/(8:l + a:.l) (s> 0) cosat 8/(S:.l + a:l) (8) 0) e at f(t) F(s +a) U(t-a)orUa(t)(a~O) e as/S (s>O) t5(t - a) (a > 0) e-as U(t - a)f(t ~ a) or Ua(t)f(t - a) f(n)(t) f * g(t) = f~ f(r)g(t - r)dr 7. (12 marks) Consider the following matrix A and its row reduced echelon fonn R (2 0 -1 1 4) A = .2 0 3 5 12 , 10125 (10013) R= 0 0 1 1 2 . o 0 000 (a) Write down a basis for the row space of A, and a basis for the column space of A.
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Unformatted text preview: (b) Find a basis for the null space (also known as the kernel) of A. (c) Find an orthogonal basis for the column space of A. MATH-263 Apri1200B 2 has an eigenvalue ). = 5 + 2i and corresponding 8. (10 marks) Find the eigenvalues and corresponding eigenvectors of the symmetric matrix A"=(2 4) 4 -4 Find an orthogonal matrix P with A = PDpT, and D diagonal. Compute the matrices in the standard basis of orthogonal projection onto each eigenspace. ( 6 -1) 9. (12 marks) The matrix A = 5 4 eigenvector ( 1 ~ 2i ) (a) Write down a bas~, in -vectorform, of real solutions of the system x; =6XI-X2 x~ = 5XI + 4X2 (b) Find functions XI(t) and X2(t) which satisfy the system above as well as the initial conditions Xl (0) = 1, X2(0) = O. (c) How do solutions of the system behave as t -t-00 ?...
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April 2008 - (b) Find a basis for the null space (also...

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