4) Given the first order equation x'= [lx-x where x = x(t) and u is a...
III) Consider the following differential equation: 5•(t) + 3x(t) - 4 = 0, x(0) = 2. 1. Find the backward solution. 2. Is this solution convergent or divergent? Justify your answer. 3. Determine the stationary solution and indicate whether it is stable or unstable. 4. Sketch a phase diagram and a time-path diagram.
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,...
Nonlinear differential equations and Bifurcation theory. Given the ordinary differential equation =1+re+ where the parameter r is a given real number. (a) Sketch all the qualitatively different vector fields that occur as r is varied. (b) Show that a saddle-node bifurcation occurs at a critical value of r, to be determined. (c) Sketch the bifurcation diagram of fixed points r* versus r.
III) Consider the following differential equation: 5i(t) + 3r(t) - 4 = 0, 7(0) = 2. 1. Find the backward solution. 1,5 mark 2. Is this solution convergent or divergent? Justify your answer. mark 3. Determine the stationary solution and indicate whether it is stable or unstable. 1 mark 4. Sketch a phase diagram and a time-path diagram. 1,5 mark
(0) is a lower- Consider the matrix equation Lx u, where L triangular square matrix and x = (p" and u = (u)' are column vectors. In view of Example 97: Solve the n equations for the n variables x1,x2, . . . , rn respectively. 1-12, . Example 97 We can find general formulas that characterize the procedure used in the previous example. Suppose we want to solve the equation Ux = v, where x = (x)' and v-(v)'...
Given the equation y' 9-16y, a) Find all Equilibrium solutions b) Determine whether each solution is stable, unstable or neither. c) Sketch the direction field. Given the equation y' 9-16y, a) Find all Equilibrium solutions b) Determine whether each solution is stable, unstable or neither. c) Sketch the direction field.
1 III) Consider the following differential equation: 5:(t)+32(t) - 4 = 0, 2(0) = 2. 1. Find the backward solution. 1,5 mark 2. Is this solution convergent or divergent? Justify your answer. mark 3. Determine the stationary solution and indicate whether it is stable or unstable. 1 mark 4. Sketch a phase diagram and a time-path diagram. 1,5 mark
x'=r (1 - 2 / 2 x where r and K are positive constants, is called the logistic equation. It is used in a number of scientific disciplines, but primarily (and historically) in population dynamics where z(t) is the size (numbers or density) of individuals in a biological population. For application to population dynamics ä(t) cannot be negative. If the solution (t) vanishes at some time, then we interpret this biologically as population extinction. (a) Draw the phase line portrait...
3. For the following example there is a bifurcation at the origin for 0. Find the evolution equation on the center manifold correct to whatever order is necessary to determine the type of bifurcation. Give a quick sketch of the 2Dflow for μ < 0 and μ > 0. 2r y - r2 [Hint: Let μ-€2, treating € as a parameter. Write down the system in (u, v, e) variables, where u, v are as in Problem 2. For this...
Find the solution to the heat equation on the infinite domain ∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,0,|x|<1|x|>1.∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,|x|<10,|x|>1. in terms of the error function. Q1 (10 points) Find the solution to the heat equation on the infinite domain azu ди at k -00<x<0, t>0, ar2 u(x,0) (X, 1x < 1 10, [] > 1. in terms of the error function. + Drag and drop your files or click to browse...