Given question solved .
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).
slove the system eqution: d^3y(t)/dt^3 - 2 d^2y(t)/dt^2 - 5 dy(t)/dt +6 y(t) = 2 d^2u(t)/dt^2 +du(t)/dt +u(t) A) compute the transfer function Y(s)/U(s)? B)Find inverse Laplace for y(t) and x(t)? C) find the final value of the system? D)find the initial value of the system? Please solve clearly with steps.
Solve for y(t). dy/dt + 2x = et dx/dt-2y= 1 +t when x(0) = 1, y(0) = 2
dt - Solve the following equation for y(t) using Fourier Transforms. dy(t) ? +2y(t) = { 'h(t) where h(t) is the Heaviside function: (0,t=0 h(t)= | 1,20 Note: the solution satisfies ly(t) >0 as t →+00.
-y-2x 2+2y de If - 1 + xy + y2 + x2 = 0 and it is known that day find all coordinate points on the curve where x = -1 and the line tangent to the curve is horizontal, or state that no such points exist.
Problem 2: Consider the DE y = f(y) = y® – 2y + H, where is a real parameter. (i) Give the steady state(s) and determine their stability. There will be different cases, depending on . Draw the phase-line diagram for each case. What is the critical value of where the bifurcation happens? (ii) Draw a bifurcation diagram, indicating stable states with solid curves and unstable states with dashed curves.
Solve the following non-homogeneous DE: –y" + 2y' – y = -eia with the initial conditions y(0) = 0 and y'(0) = 0.
Let T є L(C3) be defined by T(r, y, z)-(y-2-2c, z-2-2y,1-2y-22). (a) Is span((1,1,1)) invariant under T? (b) Is U = { ( (c) Is U = {(x, y, z) : x + y + z = 0} invariant under T? (d) Is λ 2 an eigenvalue of T? Is T-21 injective? (e) Find all eigenvectors of T associated to the eigenvalue λ =-3. 4. r, y,r-y) : x, y E C} invariant under T?
Solve the following Initial Value Problem: dy dt 2t + sect 2y y(0) = -5
dr Consider the system: = 4x – 2y dy = x + y dt (a) Determine the type of the equilibrium point at the origin. (35 points) (b) Find all straight-line solutions and draw the phase portrait for the system. (35 points) (c) What is the general solution to the system? (15 points) (d) Find the solution of the system with initial conditions: x(0) = 1 and y(0) = -1. (15 points)