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- ts) Find two linearly independent solutions of 2x²y" - xy + (2x + 1)y=0, x > 0 of the form Y₁ = x¹(1+ a₁ + a₂c² + a3x³ + ...) Y₂ = x²(1 + b₁c + b₂x² + b3x³ + ...) where T1 T2. Enter T1 T2 a1 = a₂ = a3 = b₁ b₂ TE b3 N - || || || || =6. 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? OsQ.No.4 Differentiate the following Y= (x²+3)x1 Y= ax²+b/ cx Y= 7x*+2x³-3x+374. Let f(x, y) = y³ +32²-6ry-9y-2, Find absolute maximum and minimum of this function on the triangle with vertices (0,0), (2, 2) and (2,-2). Draw the triangle. --- Y xConsider a normal, homogeneous, constant coefficient system of differential equations: yi = ay1 +A12Y2 +Which of the following functions V(y1, Y2) is a Lyapunov function for the dynamical system yi = -(y1 – y2)e(un -)* Ý2 = (y1 – y2)e(% 4)* – 2y? %3D Select one: a. V(y1, Y2) = e(y-4)² O b. V(y1, Y2) = y5 O c. V(y1, Y2) = -e-92)* + y – 1 + y – 1 d. V(y1, Y2) = elon 2)² * – y Y2Evaluate Ic1² (2x - y) dx + (x + 7y) dy. C: x-axis from (0, 0) to (0, -7) and (0, -7) to (3, -7)A laser’s beam shines along the ray given by r1(t) = (1, 2, 4) +t(2, 1, −1) for t ≥ 0. A second laser’s beam shines along the ray givenby r2(s) = (6, 3, −1) + s(−5, 2, c) for s ≥ 0, where the value of c allows for the adjustment of the z-coordinate of its direction vector. Findthe value of c that will make the two beams intersect.Problem : If f(x, y, z) = In(x³ + yl2 + 2) + V(712 + y+ 2³), then find f2, fy and f; at (1, 1, 1).