In Problems 1-12 proceed as in Example 1 to solve the given boundary-value problem.
10. A string initially at rest on the x-axis is secured on the x-axis at x = 0 and x = 1. If the string is allowed to fall under its own weight for t > 0, the displacement u(x, t) satisfies
where g is the acceleration of gravity. Solve for u(x, t).
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Differential Equations with Boundary-Value Problems (MindTap Course List)
- 8. Decide which of the following sets are linearly independent in R". Justify your answer in each case. (a) X₁ = = {(1,0), (0, 1)} ℃R² (b) X₂ = {(1,0), (2,0)} CR² (c) X3 = {(-1,0), (0,0)} ℃ R² (d) X4 = {(1,0), (0, 1), (1, 1)} C R² (e) X5 = {(1,0,0), (0, 1, 0), (1, 1, 1)} CR³ (f) X6 = {(1,0,0), (0, 1, 0), (1, 1, 1), (−1, 0, 1)} ℃ R³ (g) X7 = {(0,0,0), (0, 1, 0), (1, 1, 1), (–1, 0, 1)} ℃ R³ (h) X8 = {(0, 1,0), (1, 1, 1)} C R³ (i) X9 = {(4, 3, 0, 0), (0, 0, 1, 1), (0, 0, 0, 1), (1, 0, 0, 1), (0, 1, 0, 1)} ℃ R4 (j) X10 = {(1, 2, 0, 0), (0, 2, 3, 0), (0, 0, −1, 1)} ℃ Rª 9. Decide, for the sets X; above, for which i, j = {1, 2, ..., 10}, span(X₂) = span(X;).arrow_forward3. Find the extreme values of the function f(z, y, z) = r+y+z*, subject to the constraint r2+y+z = 1.arrow_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_forward
- 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_forward6. A flying robot is programed to follow the following three-dimensional position function while navigating an open space with no wind. The displacements in x, y, and z are measured in meters and t in seconds. x(t) = 0.0560t* – 1.200t² + 2.300t + 0.3400 y(t) = 0.6700t3 – 1.570t² z(t) = 0.2200t³ (a) At t = 1.44 s, what angle does the robot's velocity vector, v, make with its acceleration vector, å? Osarrow_forward4. Let u be a root of ƒ = t³ − t² + t + 2 € Q[t] and K = Q(u). (a) Show that f = mo(u). (b) Express (u²+u+1) (u²-u) and (u-1)-¹ in the form au²+bu+c, for some a, b, c € Q.arrow_forward2. A bug is crawling along the surface defined by x³ + y²z – z³ = 5. The bug is currently at the point (2, –2, –1). (a) If the bug moves along the surface by increasing its y-coordinate and keeping x = quickly is its z-coordinate changing? = 2, then how (b) If, instead, the bug moved from (2, –2, –1) along the surface by increasing its z-coordinate, and keeping y = -2, then how quickly is its x-coordinate changing?arrow_forward13.Let x(1)(t)=(ettet),x(2)(t)=(1t).x1t=ettet,x2t=1t. Show that x(1)(t) and x(2)(t) are linearly dependent at each point in the interval 0 ≤ t ≤ 1. Nevertheless, show that x(1)(t) and x(2)(t) are linearly independent on 0 ≤ t ≤ 1.arrow_forward2. Let f: R → R be a differentiable convex function. Consider the following problem: (P) min f(x) s.t. x ≥ 0. Let v(x), w(x): R" → R" be two vector-valued functions of x. Show using KKT conditions that TER" is an optimal solution to problem (P) if and only if is a solution to the following system: Vf(x) ≥ 0 x>0 w(x) Tv(x) = 0 In the process, find the functions v(x) and w(x). Advice: Be sure to justify the "if and only if".arrow_forward5. For F(x, y) = (2x³ + xy², x²y - 2y³) from problem 4., compute [F.Tds along the line segment from P(1, 1) to Q(2, 2) using the fundamental theorem.arrow_forward1. Find the orthogonal trajectories of the family of curve z? + 3y = cy ,where c is an arbitrary constant and which is passing through (10, 13). إضافة شرح. . . Bayan < Varrow_forward1. The maximum and the minimum values of the function f(r, y, z) = 1 + 2y +2z subject to the constraint g(r, y, z) = a +Y + - 9 = 0 are f(b, 2, e) Which of the following is true for the constants a, b, e and d? +z' 9 and (-1,a, -2) d, respectively. 4 Let D (a) a = -2, 6 = -1, c= 2. d=-9 the coo the eylir (b) (c) (d) (e) a =-2, 6-1, - -2, d=-9 a = -2, b=1, c = 2, are equi a = -2, b = 1, e=1, d= -9 a = -2, b== 1, C=2,arrow_forwardarrow_back_iosarrow_forward_ios
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