Find Fundamental Matrix for Variable Coefficient System

[Physics_wiz]

I want to find the fundamental matrix for the following system:


$$\dotx = \left(\begin{array}{cc}1/t&0\\0&1/t\end{array}\right)x + \left(\begin{array}{rr}1&1\end{array}\right)u$$

$$y = (1, 1)x$$

There's supposed to be an x' before the first equal sign but I can't get it to work for some reason. Also, all variables are matrices of appropriate dimensions.

now, the book isn't clear on how to do this when the system has variable coefficients like the one above. Wikipedia says that if the system is diagonal, we can just integrate the coefficients matrix to find the fundamental matrix. However, the professor mentioned something in class about putting unknown functions of t in the fundamental matrix then using this property:

d/dt(fundamental matrix) = A(t) * fundamental matrix

to find those functions. A(t) is the coefficients matrix. I get different solutions when I do it the wikipedia method and when I do it the professor's way. The thing is, I'm not sure if this is exactly what the professor said because he just mentioned it between the words and I didn't get a chance to write it down. Any hints?

 

[HallsofIvy]

You need a space between "\dot" and "x" so the LaTex recognizes the "\dot" command. However, that would just put the dot in front of the x, not above it. Just "x'" is sufficient.

Letting x= <u, v>, the differential equation is equivalent to the two equations
u'= (1/t) u and v'= (1/t)v. Can you find u and v?

 

[Physics_wiz]

You need a space between "\dot" and "x" so the LaTex recognizes the "\dot" command. However, that would just put the dot in front of the x, not above it. Just "x'" is sufficient.

Letting x= <u, v>, the differential equation is equivalent to the two equations
u'= (1/t) u and v'= (1/t)v. Can you find u and v?

Yes, I can easily solve this system of equations. The method asked us to solve it using the fundamental matrix though. I figured out how to do it...I can post the answer if anyone's interested.

 

[Barkan]

Yes, I can easily solve this system of equations. The method asked us to solve it using the fundamental matrix though. I figured out how to do it...I can post the answer if anyone's interested.

could you show me how to solve it thru fundamental matrix?