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Final2006summer

# Final2006summer - Final Examination Summer 2006...

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Unformatted text preview: Final Examination - Summer, 2006. Diﬀerential Equations/Linear Algebra MTH 2201 Time: 2 hours 07/06/2006 Max.Credit: 60 Answer all the questions. Make your answers precise and write legibly. Calculators are NOT allowed. 1. Consider the following system of diﬀerential equations. x y = −x + 3y = −3x + 5y (a) Write the above system in the form Y = AY where A is a 2 × 2 matrix and Y is a 2 × 1 vector. [2] (b) Find the eigenvalues and eigen vectors of the matrix A. (c) Find a solution of the system Aq = 1 1 . [1+2] [2] [3] (d) Find two linearly independent solutions of the system Y = AY . 2. Use Laplace transform methods to solve the system of diﬀerential equations x y = 4x − 2y + 2u(t − 1) = 3x − y + u(t − 1) 1 subject to the initial conditions x(0) = 0, y (0) = 2 . [10] [8] 3. Solve the IVP, using the method of variation of parameters: 4y − y = te 2 , t y (0) = 1, y (0) = 0. 4. Find the general solution of 2y − 6y = t2 using the method of undertermined coeﬃcients. 5. Solve the homogeneous diﬀerential equation (y 2 + ty ) dt + t2 dy = 0. 6. Solve the Bernoulli’s equation t2 dy − 2ty = 3y 4 . dt d d 7. Find the general solution of 16 dxy + 24 dxy + 9y = 0. 4 2 4 2 [7] [5] [5] [5] 8. Is the equation (ycosx + 2xey ) dx + (sinx + x2 ey − 1) dy = 0, exact? If it is exact, ﬁnd the general solution of the equation. [10] 1 ...
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