1) Solve the initial value problem
dx/dt = -2x+4,x(0)=5.
Sketch the solution.
2) Solve the initial value problem d2x/dt2 + dx/dt - 6x = 0, x(0) = 1, x'(0) = 1.
3) Solve
Using Gaussian elimination. Show the steps and write your answer in parametric form.
4) Determine the eigen values and corresponding eigen vectors of
5) Verify that (1,0) is one of the critical points for the nonlinear system
x'=y-x2 + 1
y'=-x-y+1
What is the other critical point? Use the linearized system at (1,0) to determine the stability of (1,0) for the nonlinear system.
1) Assume you are standing at the edge of a 30 meter building and throwa0.2kg ball upward with velocity 20m/second. If the air resistance is providing a force whose magnitude is 0.04 times the speed of the ball, then determine the maximum height of the ball (use g=9.8 in the setup) and the length of time the ball is in the air.
2) Solve the spring mass equation
2x''+2x'+3x=0, x(0)=0,x'(0)=1
For x. How many times will the mass cross equilibrium in the time interval [0,6]?
3)
Using the origin as the initial guess, find the first three iterations of the Gauss-Seidel method when solving
Check the eigen value condition to make sure convergence is guaranteed.
4) First verify that (1,1) is a critical point of the fi order system
x'= x(2-x-y)
y' = y(5-2x-3y)
Next determine the stability of the critical point for both the linearized and nonlinear system using the linearization process (that is, explicitly writed own the linearized system at (1,1) and use the eigen values of the corresponding coefficient matrix).
5) First verify that (2,1) is a critical point for the first order system
x' = 3x-3xy.
y' = -2y+xy
Next determine the linearized system at(2,1)and show that (2,1)is a stable(but not asymptotically stable) critical point for the linearized system. Finally by using the numerical solver in NumSysDE.xmcd to sketch a few trajectories of the nonlinear system starting near (2,1), you should be able to determine the stability of the critical point for the nonlinear system.