In Problems 21–24 verify that the
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Chapter 8 Solutions
A First Course in Differential Equations with Modeling Applications (MindTap Course List)
- 2. Which of the following is a general solution to the following: x²y" + xy' + (36x² - 1) y (Hint: As discussed in the lecture, use Y, only when J, and J-, are linearly dependent). A. y = c₁J₁(2x) + C₂J_1(2x) 6 B. y = C₁J₁(x) + C₂Y₁(x) 3 3 C. y = c₁₂/₁(6x) + C₂Y₁(6x) 0 D. y = c₁J₁(6x) + c₂] _1(6x) 2arrow_forward11.3 11.4: Problem 7 Find the linearization L(x, y, z) of the f(x, y, z) = 2/ x³ + y³ + z³ at the point (1, 2, 3). Answer: L(x, Y, z) =||arrow_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_forward
- 9. discuss the behavior of the dynamical system Xk+1= Axk where -0.31 (@) A = [ (b) A = ["0.3 11 1.5 0.3 (b) A = 0.3 1 3Darrow_forward1. 2. 3. Write v as a linear combination of u and w, if possible, where u = (2, 3) and w = (1, -1). (Enter your answer in terms of u and w. If not possible, enter IMPOSSIBLE.) v = (-2, -3) V = Solve for w where u = (1, 0, -1, 1) and v = (2, 0, 3, -1). w + 2v = -4u W = Write each vector as a linear combination of the vectors in S. (If not possible, enter IMPOSSIBLE.) S = {(2, -1, 3), (5, 0, 4)) (a) z = (7, -6, 14). Z= (b) v = V = (c) w = (3,-9, 15) W = (d) v = (18, - 1, 59) )$₁ U= $₁ + u = (2, 1, -1) )$₁arrow_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
- 3.16. Let P-1 and 2-11] = Q = Find a 2 x 2 matrix X such that PXQ: = -4 1] [2]arrow_forward1. Solve for x and y in xy + 8 + j(x²y + y) = 4x + 4 + j(xy² + x) A. 2, 2, B. 2,3 C. 3, 2 2. Determine the principal value of (3 + j4)¹ +² +j2 A. 0.42+j0.56 C. -0.42-j0.66, B. 0.42+j0.66 D. 0.42-j0.66 3. Using the properties of complex numbers. determine the two square roots of 3-j2 A. +1.82+j0.55, C. 1.82 + j0.55 B. +1.82±j0.55 D. +1.82 + j0.55 4. Evaluate: BE CALC 3-14 3+14 + 3+j4 3-j4 A. 2.44 +j4/ B. 2.44-j4 C. -2.44 + j4 D. 2.44 +j5 Evaluate log; (3 + j4). A. 0.6+j1.02 C. -0.6-j1.02 B. -0.6+j1.02 D. 0.6-j1.02, 6. The following three vectors are given; A = 20 +j20, B = 30/120° and C= 10+ j0, find AB/C C. 95/-50° B. 85-75% A. 70/45° D. 75/70" 7. If 100+5x/45° = 200/-e. Find x and 8. A. 24. 23.28 B. 23.28. 32.3° C. 23.28. 24.3% D. 23, 42.8° 8. Determine the principal value of cosh' (j0.5). A. In (1+j5) C. In j5 B. In (1± √5), D. In j(1 + √5) 2 5 1 = 9. In A-2B-C=0. if A= 2B-C-0. if A- and B-₁ find C |² -1 3 2 3 8 -3 8 3 91 C. A. 3 0 0 -3 -8 -8 -3 3 D. B. | 3 0 -3 10. Solve for a and b…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
- 6. 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_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_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_forward
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