i still can't understand that:
λ1 = 0 > λ2.
Line of critical points.
The critical points are not isolated –they lie on the line
through 0 with direction v1.
x = c1v1 + c2eλ2tv2
As t → ∞ x → c1v1 along a line parallel to v2.
why do the critical points lie on the line through 0 with direction v1?
MIT 18.03SC Differential Equations
Stacey Warren - Expert brainly.com
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Where is this question in the MIT courseware? Or can you completely re-produce the question?
I'm sorry I didn't make it clear.
Suppose a matrix A has real eigenvalues with two independent eigenvectors.
Let λ1, λ2 be the eigenvalues and v1 and v2 the corresponding eigenvectors.
⇒ general solution to the differential equation x'=Ax is x = c1e ^(λ1*t)v1 + c2e^(λ2*t)v2.
when λ1 = 0 > λ2, The critical points are not isolated –they lie on the line through 0 with direction v1. x = c1v1 + c2e^(λ2*t)v2 As t → ∞ x → c1v1 along a line parallel to v2. why do the critical points lie on the line through 0 with direction v1?
here is the graph of the solution to the diffenrential equation
The question is here:http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iv-first-order-systems/qualitative-behavior-phase-portraits/MIT18_03SCF11_s34_6text.pdf