Write the system in equation form?

[mathrocks]

Hi, I'm given a matrix and I need to write it in equation form so that I will have three equations, using x(t), y(t), and z(t)

The matrix is a 3x3
[3 -2 0]
x'= [-1 3 -2] *x
[0 -1 3]

I know how to rewrite it using only x(t) but I'm not sure how to do it using y(t) and z(t).

thanks!

 

[HallsofIvy]

Your problem may be that you are using " x(t)" in two different ways- as a vector function and as a component of the vector. Your
[3 -2 1]
[-1 3 2] x
[0 -1 3]

is the same as

[3 -2 1][x(t)]
[-1 3 2][y(t)]
[0 -1 3][z(t)]

Do the matrix multiplication: that's
[3x(t)- 2y(t)+ z(t)]
[-x(t)+ 3y(t)+2z(t)]
[0x(t)- y(t)+3z(t)]

so your system of differential equations is x'(t)= 3x(t)- 2y(t)+ z(t),
y'(t)= -x(t)+ 3y(t)+ 2z(t), and z'(t) -y(t)+ 3z(t).

 

[M98Ranger]

To write the system in equation form, we can use the notation x'(t) to represent the derivative of x with respect to time. Using this notation, we can rewrite the given matrix equation as follows:

x'(t) = 3x(t) - 2y(t)
y'(t) = -x(t) + 3y(t) - 2z(t)
z'(t) = -y(t) + 3z(t)

These three equations represent the system in equation form, where x(t), y(t), and z(t) are the variables and their derivatives represent the rate of change of each variable with respect to time. This notation allows us to easily see the relationships between the variables and how they change over time.