# Thread: Cauchy problem

1. ## Cauchy problem

x(with a dot above) = Ax, x(0) = (1,1,1),
where A = |-1 1 0| |0 -1 1| |1 0 -1|

Could you hint the steps I should do.

Thank you

2. Hints: In general, the solution to x'= Ax is of the form $\displaystyle x(t)= Ce^{At}$. You need to determine what e^{At} is for matrix A. Find the eigenvalues and eigenvectors (and, if necessary, the generalized eigenvectors) of the coefficient matrix. Use that to diagonalize the matrix or put it into Jordan normal form and then find $\displaystyle e^{At}$.

Alternatively, less sophisticated but simpler for a very simple equation, rewrite this as a system of equations: Writing the vector x as (x(t), y(t), z(t)), we have x'= -x+ y, y'= -y, z'= x- z.

From y'= -y, we have $\displaystyle y= Ce^{-t}$ and, since y(0)= 1, $\displaystyle y= e^{-t}$. Then $\displaystyle x'= -x+ y= -x+ e^{-t}$, a first order linear equation. Once you have solved that, put it into z'= x- z to get a linear first order equation for z.

3. Dear forum members,

I tried to do the steps showed above,

but I can not understand these:

1. According to Jordan normal form:

$\displaystyle J = P^{-1}AP$ P is column of vectors. But how can I find $\displaystyle P^{-1}$ Please can you give me the formula of it?

2. The next step is finding $\displaystyle e^{At}$ Can you explain in easier way))
Thank you very very much!