DE09springfinal

DE09springfinal - Final Examination Spring 2009...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Final Examination - Spring, 2009. Differential Equations/Linear Algebra MTH 2201 Time: 2 hours 05/05/2009 Max.Credit: 60 Answer all the questions. Make your answers precise and write legibly. Calculators are NOT allowed. 1. (a) Let a be a non-zero parameter. Evaluate the determinant a −2 2 D = −2 a −2 . 2 −2 a (b) Solve the following system of linear equations: 2x1 −2x2 +2x3 = 0 −2x1 +2x2 −2x3 = 0 2x1 −2x2 +2x3 = 0 [5] [4] (c) Find two linearly independent vectors that span the solution space of the above system. [2] (d) Solve the following system of linear equations: −4x1 −2x2 +2x3 = 0 −2x1 −4x2 −2x3 = 0 2x1 −2x2 −4x3 = 0 [3] (e) Use the results obtained in parts (a), (b), (c) and (d) and find the general solution of the following first order system of ODE: [6] x y z = x − 2y + 2z = −2x + y − 2z = 2x − 2y + z [5] [5] [5] 2. (a) Find the Laplace transform of the function f (t) = et cos t. (b) Find the inverse Laplace transform of F (s) = 1 s(s2 −2s+2) (c) Use the Laplace transform to solve the initial value problem y − y = et cos t, y (0) = 0, y (0) = 0. 3. Use the method of undertermined coefficients and find the general solution of the second order ODE [5] y + 3y + 3y + y = 30e−x . 1 4. (a) Find the general solution of the Cauchy-Euler equation: x2 y − 3xy + 3y = 0. [4] (b) Use the method of variation parameters to find a particular solution of the ODE x2 y − 3xy + 3y = 2x2 ex . [6] 5. Solve the Bernoulli equations: y = y (xy 3 − 1). [5] 6. Show that the first order ODE xy = 2xex − y + 6x2 is an exact DE and find the solution of the ODE. [5] 2 ...
View Full Document

{[ snackBarMessage ]}

Page1 / 2

DE09springfinal - Final Examination Spring 2009...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online