Classify the critical points of the plane autonomous system corresponding to the second orde differential equation ä+v(x²-1) + x = 0.
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- Explain the steps for solving a system of equations using Cramer’s rule.Solve for the orthogonal trasctories x* ( 3k-x) - ky2 = xy2(2) Sketch the direction field associated to the system I'=x²-y-3, y = y + x² - 5. Include the x-nullclines, y-nullclines, equilibria and ordinal directions in the remaining regions.
- Xn+1= a(Xn^2-Xn^3) a) being a discrete dynamic eq, what are the fixed pts of this system as a function for a ∈ [1, 5]. Decribe the stability for this fixed point as well as x ∈ [0, 1]. b) What is the possibility of having 2, 3 and 4 cycles for the system w/ x ∈ [0, 1]? Use the computer program maple to show this.find the orthogonal trjectories of x=c^y^2Find 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
- Define a new interpretation of the description “dynamical system” for a collection of interdependent differential equations?ii-Consider the nonlinear system *+ x³ + sinx = 0. Assume that V(x) = x + (x + x)² + 2f. Find f such that the origin is globally asymptotically stable.Find all critical points of the given plane autonomous system. (Enter your answers as a comma-separated list.) = x( 14 - x - ²12 v) X- (x, y) = x' y' = y(24 - y - x) 0,0, 0,24, 14,0,