TeaDrinkingGuy
New member
- Joined
- Jul 29, 2021
- Messages
- 2
Consider the dynamical system dX/dt=AX, where X=[x,y,z] and A is a 3x3 matrix. Using the exponential matrix method in the most appropriate manner you see fit, find the general solution in each of the following cases:
i) A=[math]\begin{vmatrix} 2 & 0 & -1 \\ 0 & 2 & 0 \\ -1 & 0& 2 \end{vmatrix}[/math]
ii) A=[math]\begin{vmatrix} 1 & 2 & -1 \\ 2 & 1 & 1 \\ -1 & 0& 2 \end{vmatrix}[/math]
Then, find the solutions to these similar adapted variations.
In case i) dX/dt = AX +[imath]\begin{vmatrix} 1 \\ t \\ t^2 \end{vmatrix}[/imath] and x(0)=2 y(0)=-1 z(0)=-4
In case ii) dX/dt = AX +[imath]\begin{vmatrix} 0 \\ 0 \\ e^t \end{vmatrix}[/imath] and x(0)=0 y(0)=0 z(0)=6
I know there is a lot here, but I would sincerely appreciate any help. I've been trying to crack this all day and I feel like I'm banging my head against a brick wall. Thank you in advance.
i) A=[math]\begin{vmatrix} 2 & 0 & -1 \\ 0 & 2 & 0 \\ -1 & 0& 2 \end{vmatrix}[/math]
ii) A=[math]\begin{vmatrix} 1 & 2 & -1 \\ 2 & 1 & 1 \\ -1 & 0& 2 \end{vmatrix}[/math]
Then, find the solutions to these similar adapted variations.
In case i) dX/dt = AX +[imath]\begin{vmatrix} 1 \\ t \\ t^2 \end{vmatrix}[/imath] and x(0)=2 y(0)=-1 z(0)=-4
In case ii) dX/dt = AX +[imath]\begin{vmatrix} 0 \\ 0 \\ e^t \end{vmatrix}[/imath] and x(0)=0 y(0)=0 z(0)=6
I know there is a lot here, but I would sincerely appreciate any help. I've been trying to crack this all day and I feel like I'm banging my head against a brick wall. Thank you in advance.