Question: 7. (12 Points) Solve the following initial value problem: F. u (0) 5 u- (0) -7 15 us (0) 0 1 0 It is known

7. (12 Points) Solve the following initial value problem: F. u (0) 5 u- (0) -7 15 us (0) 0 1 0 It is known that the coefficient matrix ) 0 1 has the following eigenvalues and associated 15 1 1 -3+ 40 -3-41 eigenvectors: n = 3, V, = ; = -1+2i, v2 = -5 - 101 ; and ry = -1 - 2i, v, = -5 + 10i 25 25 Your answer should be a real-valued vector function of t.8. (2 Points) Plot the function g (t) = uz (t) - us (t). (12 Points) Solve the following initial value problem: y" + 2y'+ 5y = uz(t) -us(t), y(0) =0, y'(0) =0

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