A linear system of differential equations is given by {x_1 = ax_1 + bx_2, x_2 =
ID: 3111433 • Letter: A
Question
A linear system of differential equations is given by {x_1 = ax_1 + bx_2, x_2 = cx_1 + dx_2. Writing this as a matrix equation, we have x = Ax, where A = (a b c d). Recall that the trace of a matrix is the sum of its diagonal entries, so Tr(A) = a + d and that the determinant of a 2 times 2 matrix is det(A) = ad - bc. Find conditions on Tr(A) and det(A) such that the equilibrium of (1) is a stable node, an unstable node, a stable spiral, an unstable spiral or a saddle. For instance, you should find that the equilibrium is a saddle if and only if det(A)Explanation / Answer
Unstable node is obtained if both the eigen values are positive so then sum of eigen values or Tr(A)>0 and product of eigen values or det(A)>0
Asymtotically stable node is obtained when both the eigen values are negative i.e. sum of eigen values or tr(A)<0 and product of eigen values or det(A)>0
Stable spiral is obtained if real part of the complex eigen value is negative, since complex roots come in pair so sum of eigen values or tr(A)<0 and det(A)>0
unstable spiral is obtained if the real part is positive i.e. tr(A)>0 and det(a)>0
saddle will be obtained if signs of two eigen values are opposite so their product i.e. det(A)<0
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