In Problems 21–24 verify that the
24.
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Chapter 8 Solutions
First Course in Differential Equations (Instructor's)
- Suppose 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_forward9. P = 15 -4 -7 2e31 – 8e- -4e31 + 2e- ž(1) = | 3e3t – 20e- -6e31 + 5et Show that x1 (t) is a solution to the system x = Px by evaluating derivatives and the matrix product -4 ž(1) = | 15 -7 Enter your answers in terms of the variable t. Show that x2(t) is a solution to the system x' = Px by evaluating derivatives and the matrix product 9. 3(1) = | 15 -4 X2(t) -7 Enter your answers in terms of the variable t.arrow_forward4. Find the standard matrix for T where T(a) (2x,+x 1-2x2).arrow_forward
- 14. Assume x E R. Give the matrix associated with the quadratic form 10(x,) - 5x,x2 + 6(x,).arrow_forward4. (S.10). Use Gaussian elimination with backward substitution to solve the following linear system: 2.r1 + 12 – 13 = 5, 1 + 12 – 3r3 = -9, -I1 + 12 +2r3 = 9;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
- 7. Find two linearly independent solutions of y" + 3ay = 0 of the form y₁=1+ a32³ +as+... 32=2+b₁¹+b727 +.... Enter the first few coefficients: as 11 ag= b₁ == 41 (numbers) (numbers) (numbers) ›(numbers)arrow_forward2. 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_forward1. Find the linearization of x3 − x at a = 2.arrow_forward
- 4. Verify that the given vectors of this system of ODEs are solutions, and use the Wronskian to verify that they are linearly independent. Write the general solution. e2t - (³ 5 x' = 3 -1 -3 X, X1 = 9 x2 = ( e-2t 4) 5e-2tarrow_forwardIn 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_forward2. factor 1/4, then reflects about the line y = x. 01 1/4 0 0 1/4 (a) A = (d) A = (₁ Find the standard matrix for the operator on R² which contracts with ( :) :) 1/4 0 0 1/4 (1¹4) (e) None of these (b) A = 0 1 -(;!) 0 (c) A =arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage