1. Linear Equations. Each of the following systems of ODEs involves a real parameter a. Investigate...
6. ODE Solvers ODE Initial Value Problems and Systems of ODEs The following is the van der Pol equation: y(0) = yo, y,(0) =Yo The following are solutions curves for two values of the parameter μ. Ignore the green line. Write the solution as a system of equations. Select an appropriate solver for each case, that is, for μ-1 and μ-1000, from the MATLAB list ODE23, ODE45, ODE23s, ODE113, and ODE15s. Give the type of solver and the reason for...
For each of the following equations you should locate all equilibrium points and investigate their stability properties for different values of the parameter u ER. You should then draw a summary/bifurcation diagram, with arrows, indicating the qualitative behaviour of the non-equilibrium solutions for all values of y, including at the bifurcation point u = u*. Note any bifurcations that occur and determine which of the following terms can be used to describe the nature of the bifurcations you find: transcritical,...
Consider the system of coupled ODES: x' = x - y, y = x + xy - 6y (+) (a) Find the critical points (C+, Y*) € R2 of this system. [3 marks] Hint: One critical point is (0,0) and there are two more critical points. (b) For each critical point, find the approximate linear ODE system that is valid in a small neighbourhood of it. [6 marks] (c) Find the eigenvalues of each of the linear systems found in...
1 and 2 1. Construct a table of the possible linear systems as follows: (a) The first column contains the type of the system (sink, spiral sink, source, if it has a name. (b) The second column contains the condition on the eigenvalues that corresponds (c) The third column contains a small picture of two or more possible phase por- (d) The fourth column contains x(1)-and y(a)-graphs of typical solutions indicated Hint: The most complete table contains 14 cases. Don't...
Problem 1 (Linear Systems of Equations). (a) Determine the values of a for which the follow- ing system of equations have no solution, exactly one solution, infinitely many solutions (a + 2)y + (a2-4)2 = (0-2) (b) If A = 4-1 0 a 2b a a be the augmented matrix of a linear system of equations then evaluate the values of a and b for which the linear system has no solution? exactly one solution? one parameter solution? two parameter...
1. For each of the following systems, (i) determine all critical points, (ii) determine the corresponding linear system near each critical point, and (ii) determine the eigenvalues of each linear system and the corresponding conclusion that can be inferred about the nonlinear system. (a) dz/dt x- - zy, dy/dt 3y- xy-2y (b) dr/dt r2 + y, dy/dt=y-ay
1. Identify each of the following equations as linear or nonlinear and also determine their order dy (a) y = t 3 3 (b) ( dy + (c) sin t (d) (1y) sin t ( cos2 t 1 y sin(t)= 0 (e) Int +3etdy dt (f) 2y'-y2 =e (g) y"(t2 1)y+cos(t (h) y"sin(ty)y(t21)y 0 = 0 1. Identify each of the following equations as linear or nonlinear and also determine their order dy (a) y = t 3 3 (b)...
1. For the following two systems of linear equations answer the questions 4 + x + 2xy + 2x - 6 3x + 2x + 3x3 + 3x = 11 2x + 2x + 3.5+ 2x- 9 2x + 2x+4x3+5x - 13 3x, +2, +4x3+4x-13 3x+3x+3x2+2x, -11 (1) Solve the above systems of linear equations using naive Gauss elimination (b) solve the above systems of linear equations using Gauss elimination with partial pivoting . Axb 2. For the following matrix...
Problem 1. For each of the following systems of autonomous differential equations, sketch the nullclines and find the equilibria da dy =y-x2 + 3x-2, a+ b 1000 ,2 1-7 a+b 1100 Problem 1. For each of the following systems of autonomous differential equations, sketch the nullclines and find the equilibria da dy =y-x2 + 3x-2, a+ b 1000 ,2 1-7 a+b 1100
1. For each of the following systems of linear equations, find: • the augmented matrix • the coefficient matrix • the reduced row echelon form of the augmented matrix • the rank of the augmented matrix • all solutions to the original system of equations Show your work, and use Gauss-Jordan elimination (row reduction) when finding the reduced row echelon forms. (b) 2 + 2x W 2w - 2y - y + y + 3z = 0 = 1 +...