In Problems 21–24 verify that the
23.
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Chapter 8 Solutions
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- In Problem. -24, the given vector functions are solutions to a system x' (t) = Ax(t). Determine whether they form a fundamental solution set. If they do, find a fundamental matrix for the system and give a general solution. -E]. et 24. x₁ = X2 sint COS [ -sint_ X cost sin/ COSTarrow_forwardSuppose a, = (1 -1 1 1) 5. az = (1 0 1 0) %3D %3D az = (1 1 1 1)''a¸ = (3 2 3 0)" - Please find out whether a1,a2,a3,&4 are linear independent or not?arrow_forward10. Determine three linearly independent solutions to the equation y" + 2y" – 3y = 0 of the form y(x) = e"*, where r is a real number. Remember to prove that these solutions are indeed linearly independent.arrow_forward
- Solve this.arrow_forward9. discuss the behavior of the dynamical system xk+1= AXk where -0.3] 2 ] (b) A = 1.5 (a) A = 1.2 0.3 0.41 -0.3 1.1arrow_forward6. Find the general solution of the system of differential equations -B -B X = dt -- -B х, -B -B where a 0 and B 0 are real nonzero constants. Hint: The characteristic polynomial of the coefficient matrix is -(A– a - B)²(A – a + 23).arrow_forward
- 3 [₁³e- t³e-2t dtarrow_forwardTr.28.arrow_forward10. Find the general solution of the system of differential equations 3 -2 -2 d. X = -3 -2 -6 X dt 3 10 1 + 2tet + 3t?et + 4t°et 3 1 -3 Hint: The characteristic polymomial of the coefficient matrix is -(A- 4)²(A- 3). Moreover (:) 2 1 Xp(t) = t²et +t³et +t'e3t -1 -1 -3 is a particular solution of the system.arrow_forward
- 2. Given the following 2 x 2 linear system with constant coefficients x' = Ax (H) x= Ax+g(t), (N) where g is not the zero vector. Which of the following statements are true? Justify your answers. A. If , is a solution to (H) and 7, is a solution to (N), then , +27, is a solution to (N). B. If , and 2 are both solutions to (N), then ₁-2 is a solution to (H).arrow_forward6. Find the general solution of the system of differen- tial equations 6 -1 1 1 d 6t X - e 6. -1 1 2 dt 1 1 3 Hint: The characteristic polynomial of the coefficient matrix is –(A – 7)²(A – 4). | 2.arrow_forwardQ.3 Find the first foster form of the driving point function of Z(S) = 2 (S+2) (S+5)/(S+4)(S+6)arrow_forward
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