Thread: sytem of diff eq View Single Post

## sytem of diff eq

This is a given system: $$D\vec{y} = A\vec{y} + \vec{b}$$
With $$A=\left\begin{array}{ccc}1&1&1\\0&2&1\\0&0&3\end{array}\right$$
And $$\vec{b}=\left\begin{array}{c}e^4^t\\0\\0\end{array}\right$$

We find $$\vec{y}_H = Y(t) \cdot \vec{c}$$

With $$Y(t)=\left\begin{array}{ccc}e^t&e^2^t&e^3^t\\0&e^2^t&e^3^t\\0&0&e^3^t\e nd{array}\right$$

Since $$\vec{y} = \vec{y}_H + \vec{y}_P$$ we still need to find is $$\vec{y}_P = Y(t) \cdot \vec{c}(t)$$
$$Y(t)$$ is already known, so whe have to find $$\vec{c}(t)$$
We know that $$D\vec{c}(t)=Y^-^1(t) \cdot \vec{b}$$

Now we are gonna replace $$Y^-^1(t)$$ by $$e^A^(^-^t^)$$

With $$e^A^(^-^t^)$$ being $$e^A^t= Y(t) \cdot Y^-^1(0)$$
So we get $$D\vec{c}(t)= e^A^(^-^t^) \cdot \vec{b}$$

My question actually is why we can replace $$Y^-^1(t)$$ by $$e^A^(^-^t^)$$ since $$e^A^t^= Y(t) \cdot Y^-^1(0)$$?

P.S.: I'll post the whole excersise later if it's necessary, but I had to much trouble with the Latex code for now ;)

 PhysOrg.com science news on PhysOrg.com >> Hong Kong launches first electric taxis>> Morocco to harness the wind in energy hunt>> Galaxy's Ring of Fire