Solve Linear System of ODEs | x(0)=x0,y(0)=y0,z(0)=z0

In summary, this conversation discusses a linear system of ODE with a specific matrix. The conversation goes on to ask for analysis and solutions regarding the stability, fixed points, and behavior of solutions to the system. However, due to academic honesty policies, the conversation is closed until the student can provide contact information for their professor and get approval for seeking outside help with this problem.
  • #1
ZiniaDuttaGupta
3
0
I have another question so please help me. here we go --

Consider the following linear system of ODE :

X’ = -x – y
Y’ = x + 3y
Z’ = 4x + 6y - z

Note that the matrix of this system is exactly the same as
A = [ 1 -1 0
1 3 0
4 6 -1 ]

(a) Study the stability of the fixed point (0,0): is it a source (all solutions diverge to ∞ from it), sink (all solutions converge to it), saddle (any solution is either convergent to the fixed point or diverges to ∞), or neither?

(b) Determine stable and unstable subspaces of (0; 0). (The final answer should be: the stable subspace is spanned by vectors ..., or the stable subspace does not exist.)

(c) Draw a phase portrait of your system in the unstable subspace.

(d) Briefly describe the behaviour of solutions to this system. (e.g. "all the solutions except those in xy-plane will go to ∞ while rotating around z-axis; the solutions that start in xy-plane will stay in that plane and will rotate on the circle centered at the fixed point (0,0) - 5pts. bonus if you can give me a simple matrix of such a system!)

(e) Write down the general solution of the system above using the initial data

x(0) = x0; y(0) = y0; z(0) = z0:
 
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  • #2
Hello ZiniaDuttaGupta,

In part d) of this question, it is stated:

"5pts. bonus if you can give me a simple matrix of such a system!"

This indicates that this question is part of an assignment that contributes to your final grade, and MHB policy is to not knowingly help with graded assignments, which your professor certainly expects to be your work.

Academic honesty is taken very seriously here, and we do not wish to go against the policies of educators by giving help with problems which are meant as assessments of the abilities of their students.

I am directed by policy to close this topic until you can provide me with the contact information for your professor and he/she states in the following correspondence that it is okay for you to seek outside help with this problem.

Best Regards,

Mark.
 

Related to Solve Linear System of ODEs | x(0)=x0,y(0)=y0,z(0)=z0

1. What is a linear system of ODEs?

A linear system of ODEs (ordinary differential equations) is a set of differential equations that can be written in the form of Ax = b, where A is a matrix, x is the vector of unknown functions, and b is a vector of known functions. In other words, the equations are linearly related to each other.

2. What does x(0)=x0, y(0)=y0, z(0)=z0 mean in the context of solving a linear system of ODEs?

These are initial conditions for the unknown functions x, y, and z, respectively. They represent the values of these functions at the starting point, or initial time, of the system. In order to solve the system, we need to know these initial values.

3. How do you solve a linear system of ODEs?

There are various methods for solving a linear system of ODEs, including elimination, substitution, and matrix methods. One common approach is to use the method of undetermined coefficients, where we assume a particular form for the solution and then solve for the coefficients. Another method is to use the Laplace transform, which can simplify the equations and make them easier to solve.

4. What is the importance of solving a linear system of ODEs?

Linear systems of ODEs are used to model many real-world phenomena, such as population growth, chemical reactions, and electrical circuits. By solving these systems, we can gain insights into the behavior and evolution of these systems over time. They are also important in many areas of engineering and physics, as they provide a mathematical framework for understanding and predicting complex systems.

5. Are there any applications of solving linear systems of ODEs?

Yes, there are many applications of solving linear systems of ODEs. These include predicting the spread of infectious diseases, understanding the dynamics of predator-prey relationships, and designing control systems for robots and other machines. They are also used in fields such as economics, biology, and climate science to model and analyze complex systems.

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