- #1
motherh
- 27
- 0
A question I am doing hints that the solution (y,[itex]\dot{y}[/itex]) = (0,0) of [itex]\ddot{y}[/itex] - [itex]\frac{2}{t}[/itex][itex]\dot{y}[/itex] + y = 0 is unstable. I believe (although I am not 100% sure) that is true however I am struggling to prove it.
I can rewrite the equation as a system of equations in matrix form to get
[itex]\dot{x}[/itex] = Ax + B(t)x,
where A = [{0,1},{-1,0}], B(t) = [{0,0},{0,[itex]\frac{2}{t}[/itex]}].
This the form of all the theorems I appear to have. But all my theorems require me to find an eigenvalue of A with positive real part - which I can't here.
So basically have I made a mistake already or is there another theorem anybody knows of that can tell me the trivial solution is unstable?
I can rewrite the equation as a system of equations in matrix form to get
[itex]\dot{x}[/itex] = Ax + B(t)x,
where A = [{0,1},{-1,0}], B(t) = [{0,0},{0,[itex]\frac{2}{t}[/itex]}].
This the form of all the theorems I appear to have. But all my theorems require me to find an eigenvalue of A with positive real part - which I can't here.
So basically have I made a mistake already or is there another theorem anybody knows of that can tell me the trivial solution is unstable?