- #1
jimmycricket
- 116
- 2
Given the matrix [itex]b=\begin{pmatrix}-1&0&-1\\-4&3&-1\\0&0&-2\end{pmatrix}[/itex] decide if the system of ODEs, [itex]\frac{dx}{dt}=Bx[/itex] is decoupled. If yes find the general solution x=xh(t)
I would say the matrix is decoupled since the second equation involving [itex]2x[/itex]2(t) can be solved without the other two equations. Then the third equation can be solved without knowing [itex]x[/itex]1(t). We have:
[itex]
x'_1 = -x_1 - x_3 \\
x'_2 = -4x_1 + 3x_2 - x_3 \\
x'_3 = -2x_3
[/itex]
Im not sure where to go from here.
Homework Equations
The Attempt at a Solution
I would say the matrix is decoupled since the second equation involving [itex]2x[/itex]2(t) can be solved without the other two equations. Then the third equation can be solved without knowing [itex]x[/itex]1(t). We have:
[itex]
x'_1 = -x_1 - x_3 \\
x'_2 = -4x_1 + 3x_2 - x_3 \\
x'_3 = -2x_3
[/itex]
Im not sure where to go from here.