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Unformatted text preview: Differential Equations Math 308, Spring 2008 Texas A&M University
c F. Dos Reis Sections 9.1, 9.3
Exercise 1. Express the given systems of differential equation in matrix notation. x (t) = 7x + 2y y (t) = 3x - 2y x1 (t) = x1 - x2 + x3 - x4 x2 (t) = x1 + x4 x3 (t) = tx1 + sin(t)x3 x4 (t) = 0 Exercise 2. Express the given differential equations as matrix systems in normal form. (1 - t2 )y (t) - 2ty (t) + 2y(t) = 0 y + 2ty - 3y + t2 y = 0 Recall: Determinant: Let A be an n n matrix. Te following statements are equivalent: A is inversible the determinant of A is not 0. the only solution to AX = 0 is X = 0. The columnum (row) form a linearly independant set. 1 t e3t e2t e3t 1 t and, of B(t) = 3 e3t 0 1 9 e3t 0 0 Exercise 3. Compute the determinants of A(t) = Definition: Let A(t) be a matrix such that the entries are functions of t. A(t) is continuous if all the entries are continuous. A(t) is differentiable if all the entries (ai,j (t)) are differentiable. In this case A (t) is the matrix with entries a (t). ij Theorem: Let A, B be 2 matrix and C be a matrix with constant entries. dA d (CA) = C dt dt d dA dB (A + B) = + dt dt dt dA dB d (AB) = B+A dt dt dt 1 Differential Equations Math 308, Spring 2008 Exercise 4. Find the derivative of 1. sin t X(t) = cos 2t ln t t cos t 2 3t 1 0 3 et Texas A&M University
c F. Dos Reis 1 e2t 2. AB(t) with A(t) = t e3t and B(t) = 5 sin t e3t Exercise 5. Show that X1 (t) = 0 and X2 (t) = e3 system 1 -2 -2 1 X = 2 2 Exercise 6. Is X(t) = e-t e5t -t 2 e5t -3e - e3t e3 are solution to the homogeneous 0 2 2 1 2 1 ? 9 2 solution to X = 2 ...
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