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Problem 5 (40 pts). Given the system of nonlinear differential equations Se=y+ 2(x2 + y2 -...
Consider the nonlinear system ?x′ = ln(y^2 − x) and y'=x-y-1 (a)Find all the critical points (b)Find the corresponding linearized system near the critical points. (c) Classify the (i) type (node, saddle point, · · · ), and (ii) stability of the critical points for the corresponding linearized system. (d) What conclusion can you obtain for the type and stability of the critical points for the original nonlinear system?
Solve the following differential equations. 10. Solve the following differential equations. (a) (x2 - y2) 2 = ry (c) y" – y' cot = cot x (d) - 2y = 23
Consider the given system di = 2x²y – 3x2 - 25 y, y=-2xy? + bxy. x Incorrect (a) Determine all critical points of the given system of equations. Write your points in ascending order of their x-coordinates: if two points have the same x-coordinate, write them in ascending order of (x2.72) =( x Incorrect. (b) Find the corresponding linear system near each critical point. 1. The linear system near the critical point ($1.91) () = A (s) where: 1. The...
Problem 2- System Classification: Linearity (20pts) Circle all nonlinear terms (if any) in the following differential equations: (assume variables on left are outputs, at right are inputs) y'(t) *,4x, +4x, cos(x2) e. Problem 2- System Classification: Linearity (20pts) Circle all nonlinear terms (if any) in the following differential equations: (assume variables on left are outputs, at right are inputs) y'(t) *,4x, +4x, cos(x2) e.
Consider the system: x' = y(1 + 2x) y' = x + x2 - y2 a. Find all the equilibrium points, and linearize the system about each equilibrium point to find the type of the equilibrium point. b. Show that the system is a gradient system, and conclude that it has no periodic solutions. c. Sketch the phase portrait. Explain how you determined what the phase portrait looks like.
(1 point) Consider the system of higher order differential equations 2 Rewrite the given system of two second order differential equations as a system of four first order linear differential equations of the formy - P(t)y + g(t). Use the following change of variables y (t) y2(t)y'(t) 3 (t) y(t) у(t) z(t) -y2 4 (1 point) Consider the system of higher order differential equations 2 Rewrite the given system of two second order differential equations as a system of four...
#10 all parts In each of Problems 5 through 18: (a) Determine all critical points of the given system of equations. (b) Find the corresponding linear system near each critical point. (c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system? (d) Draw a phase portrait of the nonlinear system to confirm your conclusions or to extend them in those cases where the linear system does not provide definite information about the...
Consider the nonlinear System of differential equations di dt dt (a) Determine all critical points of the system (b) For each critical point with nonnegative x value (20) i. Determine the linearised system and discuss whether it can be used to approximate the ii. For each critical point where the approximation is valid, determine the general solution of iii. Sketch by hand the phase portrait of each linearised system where the approximation behaviour of the non-linear system the linearised system...
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each of the regions b. Linearize the system around the equilibrium point, and use your result to classify the type of the c. Use the information from parts a and b to sketch the phase portrait of the system. 2. Sketch the phase portraits for the following systems...
Problem Two: (Based on Chapra, Problems 12.9 Consider the simultaneous nonlinear equations: 2-5-y y+i- 1. Plot the equations and identify the solution graphically. Page 1 of 2 2. Solve the system of equations using successive substitution, starting with the initial guess xo-y-1.5. Show two complete iterations. Evaluate &s for the second iteration. 3. Redo Part 2 using Newton-Raphson method . Automate the solutions in Parts 2 and 3 using MATLAB scripts 5. Solve the system of nonlinear equations by calling...