Rewrite the following second order differential equations as a system of first-order equations: (a) v'' + 3v' + 5v = 0 (b) v'' + 6v' + 9v = sin(2t)
Solve the following linear systems of differential equations. Provide a sketch the phase portrait and classify the stability of the equilibrium solution as either stable, unstable or unstable saddle. (a) dx1/dt = x1 - x2 dx2/dt = 2x1 + 4x2 (c) x' =[ -1 1 -4 -3 ]x (b) dx1/dt = x1 + 2x2 dx2/dt = 4x1 + 3x2 (d) x' =[ 6 -1 5 2 ]x
Solve the following IVPs. (a) dx1/dt = -2x1 + x2 dx2/dt = 0x1 - 3x2 , X(0) = [1 1] (b) dx1/dt = -2x1 - 2x2 dx2/dt = 2x1 - 6x2 , X(0) = [-4 1].
A drug is administered to a person in a single dose. We assume that the drug dose not accumulate in body tissue, but is secreted through urine. We denote the amount of the drug in the body at time t by x1(t) and in the urine at time t by x2(t). Initially, x1(0) = 1/2 and x2(0) = 0.We describe the movement of the drug between the body and the urine by dx1/dt = -ax1(t) dx2/dt = ax1(t) a) Explain the model in words (explain each differential equation then explain it as a system). (b) Solve the linear system. (c) Describe the long term progression of the drug (Take the limits of both solutions x1(t) and x2(t) and describe what happens).
Deep in the redwood forests in California, dusky-footed rats provide up to 80% of the diet for the spotted owl, the main predator of the wood rat. So, let O(t) and R(t) stand for the owl population and the rat population at time t in months respectively. Consider the linear system dR/dt = .1R - p.O dO/dt = .4R - .5O The term -p