Matrices and general solutions

In summary, the eigenvalues for this differential equation are 1, +5i, -5i, and the general solution found is -7.
  • #1
58
0
I'm stuck on this problem that involves drawing a phase portrait but I'm at a stand still.

Find the general solution to the following differential equation:
[1 -12 -14]
x'= [1 2 -3]*x
[1 1 -2]

the eigeinvalues that I found are 1 , +5i, -5i

the general solution that I found is

c1e^e[25] + c2[cos5t+5sin5t] + c2[-5cos5t+sin5t] + c3[2] + c3t[1-5i]
[-7] [cos5t ] [sin5t ] [2] [1 ]
[6 ] [cos5t ] [sin5t ] [1] [1 ]

Am I suppose to have imaginary values in my solution?
I'm supposed to find the solution if x(0)=2i-5j+3k
I just plug 0 in for all the t's and solve for the constants which is no problem for me. The problem that I'm having is the fact that I have an imaginary value in the solution.

I'm also suppose to find the phase portrait.
 
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  • #2
No, since this problem has only real entries, you should not have any complex numbers in your solution- and I don't see why you should. If your eigenvalues are 1, 5i, and -5i, then your solutions will involve ex, cos(5x), and sin(5x).

I have trouble understanding what you mean by
c1e^e[25] + c2[cos5t+5sin5t] + c2[-5cos5t+sin5t] + c3[2] + c3t[1-5i]
[-7] [cos5t ] [sin5t ] [2] [1 ]
[6 ] [cos5t ] [sin5t ] [1] [1 ]
Is that the "vector" solution? Is [-7][cos5t][sin5t][2][1] 5 separate functions or is that a product? Each row should be a single function like your first row.
 
  • #3
It's the vector solution. I just realized that this message board gets rid of the extra spaces. 25, -7, 6 is one column matrix. cos5t+5sin5t, cos5t, cos5t is one column matrix. -5cos55t+sin5t, sin5t, sin5t is one column matrix. 2, 2, 1 is one column matrix. 1-5i, 1, 1 is one column matrix. And that's what I thought...about the imaginary values. How should look in the general solution?
 

1. What is a matrix?

A matrix is a rectangular array of numbers or expressions arranged in rows and columns. It is commonly used in mathematics, physics, computer science, and other fields to represent and manipulate data.

2. What is the purpose of matrices?

Matrices are used to solve systems of linear equations, perform transformations, and represent data in a concise and organized manner. They also have applications in computer graphics, cryptography, and optimization problems.

3. What is a general solution of a matrix?

A general solution of a matrix is a set of values that satisfy the system of equations represented by the matrix. It is often expressed in terms of variables, allowing for infinite solutions that satisfy the equation.

4. How do you find the general solution of a matrix?

To find the general solution of a matrix, you can use techniques such as row reduction, Gaussian elimination, or inverse matrices. These methods involve manipulating the matrix to reduce it to a simpler form, making it easier to solve for the variables.

5. Can matrices have more than one general solution?

Yes, matrices can have infinite solutions, meaning there can be more than one general solution. This is often the case when the matrix has more variables than equations, allowing for a range of values that satisfy the system of equations.

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