Prove that the system d dx(t) = −x+4y d = does not have periodic solutions by determining the values of a,n,m so that V (x,y) = xay" is a global Lyapunov function.

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 - || || || || =

Find the quadratic function f (x) = ax2 + bx + c that goes through (0, 1) and has a local minimum at (1,-1).

Find two linearly independent solutions of 2a?y" – xy' + (5x + 1)y= 0, x > 0 of the form Y1 = x" (1+ a1r + a2x? + a3x³+..) Y2 = x" (1+ bịT + bzx² + b3x³+..) where r1 > r2- Enter T1 a1 a2 a3 r2 = %3D b2 b3

Suppose a firm's technology is represented by the production function f(L, K) = min(L, 2K), where L is the quantity of labor and K is the quantity of capital. Consider the isoquant through input combination (L, K) = (4, 8). Which of the following statements is true? The isoquant through input combination (L, K) = (4, 8) has a corner at A input combination (L, K) = (4, 8). The isoquant through input combination (L, K) = (4, 8) has a corner at input combination (L, K) = (4, 2). %3D %3D The isoquant through input combination (L, K) = (4, 8) has a corner at C input combination (L, K) = (8, 8). The isoquant through input combination (L, K) = (4, 8) has a corner at input combination (L, K) = (8, 4).

y" -y - 2y = 0. Assume that y, (t) = e and y, (t) = et form a fundamental set of solutions. (b) Let Y3 (1) = -4e2 4 (t) = y1 (t) +4y2 (?) ys (t) = 4y, (t) – 4 y3 (t). Are y3 (t), y4 (t), and ys (t) also solutions of the given differential equation? O Yes O No O Impossible to tell

Y₁ = 1+ a3x³ + A6㺠+... Y₂ = x + b₁x¹ +b7x7 +... Enter the first few coefficients: Az = a6 = b₁ b7 ) Find two linearly independent solutions of y" + 6xy = = - 0 of the form

Let f1(x) and f2(x) are linearly independent solutions of y''+ P(x)y'+ Q(x)y = 0. Assume that f1(x) has at least two zeros (roots). Show that f2(x) has one and only one zero between consecutive zeros of f1(x)

b) The equations r² + y²s + yt² = 13 and y² + x²s+at² = 9 defines and t as implicit əs at that point. functions of r and y near the point (x,y,s,t) = (1,2,1, - 2). Find ду

Which of the following functions V(y1, y2) is a Lyapunov function for the dynamical system yi = -2yı vže'ii – yı, ý½ = -2y¡y½eisi Select one: · a. V(y1, y2) = –yı ¥2ei»% – }y? O b. V(y1,y2) = erisi + – 1 O c. = -ei + = e%i¥2 + y? Ос. V(y1, y2) O d. V(y1, y2)