1 Point For The System Of Differential Equations X T X Y 2xy Y T 8x 2y Xy The Critical Point Xo 1 (76.91 KiB) Viewed 438 times
(1 point) For the system of differential equations x' (t) = − ²x + y + 2xy y' (t) = -¹8x + 2y-xy the critical point (xo, Yo) with x > 0, yo > 0 is xo = , Yo = Expressing this system as x' = f(x, y), y = g(x, y), the Jacobian matrix at x, y is J(x, y) = [fx(x, y) fy(x, y) gx(x, y) gy(x, y)] = and the Jacobian matrix at the critical point (xo, yo) is J(xo, Yo) = The eigenvalues of this matrix are A1 <^₂ = The critical point (xo, yo) is best described as a saddle sink / stable node source / unstable node saddle center point / ellipses spiral source spiral sink none of these
(1 point) For the system of differential equations x' (t) = − ²x + y + 2xy 18 y (t) = -x +2y-xy 5 the critical point (xo, yo) with x > 0, yo > 0 is xo = , Yo = Change variables in the system by letting x(t) = x₁ + u(t), y(t) = yo + v(t). The system for u, v is u' = v' = Use u and u for the two functions, rather than u(t) and v(t) For the u, v system, the Jacobian matrix at the origin is -| You should note that this matrix is the same as J(xo, Yo) from the previous problem. A:
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