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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   (c) Find two linearly independent vectors that span the solution space of the above system.  (d) Solve the following system of linear equations: −4x1 −2x2 +2x3 = 0 −2x1 −4x2 −2x3 = 0 2x1 −2x2 −4x3 = 0  (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:  x y z = x − 2y + 2z = −2x + y − 2z = 2x − 2y + z    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  y + 3y + 3y + y = 30e−x . 1 4. (a) Find the general solution of the Cauchy-Euler equation: x2 y − 3xy + 3y = 0.  (b) Use the method of variation parameters to ﬁnd a particular solution of the ODE x2 y − 3xy + 3y = 2x2 ex .  5. Solve the Bernoulli equations: y = y (xy 3 − 1).  6. Show that the ﬁrst order ODE xy = 2xex − y + 6x2 is an exact DE and ﬁnd the solution of the ODE.  2 ...
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