Problem 3. Consider the following continuous differential equation dx dt = αx − 2xy dy dt = 3xy − y
3a (5 pts): Find the steady states of the system. 3b (15 pts): Linearize the model about each of the fixed points and determine the type of stability.
3b (15 pts): Draw the phase portrait for this system, including nullclines, flow trajectories, and all fixed points.
Problem 2 (25 pts): Two-dimensional linear ODEs For the following linear systems, identify the type of fixed point (stable node, unstable node, stable focus, unstable focus, center, or saddle point), and make a rough sketch of the solution x(t) on the axes provided on the next page. Note that your sketch only needs to be qualitatively (not quantitatively) correct.
a) [5 pts] dx dt = −x dy dt = x − 2y
b) [5 pts] dx dt = −y dy dt = x
c) [5 pts] dx dt = −y dy dt = −10x − 2y
d) [5 pts] dx dt = x dy dt = 3x − y
e) [5 pts] dx dt = x − y dy dt = 2x + y
since the question was extremely long I have tried to explain each of the parts precisely. If you cannot understand any portion feel free to ask.
Problem 3. Consider the following continuous differential equation dx dt = αx − 2xy dy dt...
dx Consider the system 2 - NICO ху 2 22 dy dt = 2y – 1- 2XY dt 2 (a) Identify all critical points of the system. (b) For each critical point, use eigenvalues to classify the critical points according to stability (stable, unstable, asymptotically stable) and type (saddle, proper node, etc).
Consider the following system: dx/dt=y(x^2+y^2-1) dy/dt= -x(x^2 +y^2-1) Find the equilibrium solution. 13. Consider the following system dx dy (e) Find the equilibrium solutions (0 Use Maple to sketch a phase portrait (me to understand the qualitative behavior of 13. Consider the following system dx dy (e) Find the equilibrium solutions (0 Use Maple to sketch a phase portrait (me to understand the qualitative behavior of
(1 point) Consider the system of differential equations dx dt = -1.6x + 0.5y, dy dt = 2.5x – 3.6y. For this system, the smaller eigenvalue is -41/10 and the larger eigenvalue is -11/10 [Note-- you may want to view a phase plane plot (right click to open in a new window).] If y' Ay is a differential equation, how would the solution curves behave? All of the solutions curves would converge towards 0. (Stable node) All of the solution...
Classify the critical point (0, 0) of the given linear system. Draw a phase portrait. dx/df 3x+ y a. dx/dt -x+ 2y dx/dt =-x +3y dy/dt -2x + y dy/dt x+ y Classify the stationary point (0, 0) of the given linear system. Draw a phase portrait. dy/dt -x+y b. dx/dt =-2x-y dx/dt-2x +5/7 y dx/dt 3x-y dx/dt 3x dy/dt 3x- y dy/dt 7x- 3y dy/dt x+y dy/dt 3y
dy -X dx2 dt =2y-x dt 2. Consider the following system of equations: phase plane, showing only the first quadrant. (a) Graph the nullclines on a (b) Find the fixed points (there are two) to determine the nature of each fixed point (i.e., source, sink, saddle, and (c) Use Jacobian analysis whether it is a node or spiral). (d) Draw the flow arrows in each region of your phase plane from part (a). You may use a computer to help...
Consider the following differential equation system: x' = 16x + 8y y = -24x – 12y (a) Find the general solution. (b) Without a computer, sketch a phase diagram that shows four linear solution trajectories and that shows one solution trajectory in each of the four regions between the separatrices. (c) Determine the solution that satisfies x(0) = 1 and y(0) = 0. x(t) = yt) = (d) The point (0,0) is a ... Osaddle point stable node unstable node...
Problem 3: Study the planar system x' = (y – 1)(y – 2), - . Find and sketch the nullclines and give the vectorfield on the nullelines. Also find all steady states and perform a linear stability analysis for each of them to find whether they are stable or unstable and give the type (saddle, node, focus).
Find the time constant t of the following differential equation: a(dy/dt)+by+cx=e(dx/dt)+f(dy/dt)+g, of the given that x is the inout, y is the output, and a through g are constants. 13, Find the time constant τ from the following differential equation, dt dt given that x is the input, y is the output and, a through g are constants. It is known that for a first-order instrument with differential equation a time constant r- alao dy the 13, Find the time...
Question 3 Consider the following linear system of differential equations dx: = 2x-3y dt dy dt (a) Write this system of differential equations in matrix form (b) Find the general solution of the system (c) Solve the initial value problem given (0) 3 and y(0)-4 (d) Verify the calculations with MATLAB Question 3 Consider the following linear system of differential equations dx: = 2x-3y dt dy dt (a) Write this system of differential equations in matrix form (b) Find the...
help with this linear algebra question dy =1 +1 dt X-- Solve the system: 5.) dx dy 2y=e + dt d