1. 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. x' = [ 1 1 -4 -3 ]x.
Non-homogeneous 1st-order Systems: Solve the following linear systems of differential equations. (a) x
Newton’s Cooling: Professor Westin lives in a two story house. One cold winter night the Professor’s furnace (which is housed on the first floor) fails! Suppose each floor exchanges heat with it’s environment according to Newton’ Law of Cooling. Suppose the system discussed in class is a model for the temperature x1 downstairs and the temperature x2 upstairs: x’ = [ x1’ x2’ ] = 1/10 [ -7 5 5 -6 ] x (a) Determine x(t) for times t > 0 if the temperatures at t = 0 are given by x = [ 70 60 ]. Use desmos to graph the components, x1(t) and x2(t), of your solution. (b) Again, determine x(t) for times t > 0 if the temperatures at t = 0 are given by x = [ 70 60 ], but now assume the furnace does not simply die in an instant, but instead dies a slow death, supplying heat downstairs at the rate e -t . That is, solve the nonhomogeneous system with f(t) = [ e^-t 0 ]. Use desmos to graph the components, x1(t) and x2(t), of your solution.
Radioactive Decay Series: Let x(t), y(t), and z(t) denote amounts of substances X, Y , and Z respectively, remaining at time t (X decays to Y and Y decays to Z where Z is the stable substance that we get at the end). A linear system of differential equations describing this dynamics can be written as dx/dt = -.12x dy/dt = .12x - .5y dz/dt = .5y (a) Explain the model in words (explain each differential equation then explain it as a system). (b) Solve for the amount of each substance if x(0) = x0 (some initial quantity), y(0) = 0, and z(0) = 0.