- #1
Kamekui
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Homework Statement
Consider the second order equation x''+ ax' + bx = 0:
(a) Convert the equation to a 2 x 2 system.
(b) Compute the eigenvalues.
(c) In circuit and spring problems, both constants are nonnegative. Assume that they
are actually positive, and show that the eigenvalues have a negative real part, and conclude
that the trivial solution is asymptotically stable.
Homework Equations
The Attempt at a Solution
(a). Let x1=x, x2=x'
x1'=x2, x2'=-ax'-b''
{{0,1},{-b,-a}}=A
(b) det(A-λI)=
\begin{bmatrix}
-λ & 1 \\
-b & -a-λ\\
\end{bmatrix}
= x2+aλ+b
→λ=-a/2+-√(a2-4b)/2
-a/2 is real and negative... but I just don't feel like I'm going about getting the solution the correct way.
(c) No idea...
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