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c31 - 18.03 Class 31 First order systems Introduction[1...

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18.03 Class 31, April 28, 2006 First order systems: Introduction [1] There are two fields in which rabbits are breeding like rabbits. Field 1 contains x(t) rabbits, field 2 contains y(t) rabbits. In both fields the rabbits breed at a rate of 3 rabbits per rabbit per year. Note that the rabbits cancel, so the units are (year)^{-1} . They can also leap over the hedge between the fields. The grass is greener in field 2, so rabbits from field 1 jump at the rate of 5 yr^{-1} , while rabbits from field 2 jump only at the rate of 1 yr^{-1}. So the equations are x' = 3x - 5x + y = -2x + y (1) y' = 3y - y + 5x = 5x + 2y (2) The net growth rate of the field 1 population is -2 because of all the jumping, and the net growth rate in field 2 is 2. On the other hand, each derivative is increased by virtue of the influx from the other field. Each of the four coefficients has a clear interpretation. This is a linear SYSTEM of equations, homogeneous. The general case looks like x' = ax + by y' = cx + dy It seems to be impossible to solve, since you need to know y to solve for x and you need to know x to solve for y. We can solve, though, by a process called ELIMINATION: use (1) to express y in terms of x: y = x' + 2x and then plug this into (2): x" + 2x' = 5x + 2(x' + 2x) or x" - 9x = 0 This is a SECOND ORDER ODE , which we can solve: The characteristic polynomial is s^2 = 9 , and the roots are +-3. We get two basic solutions, x_1 = e^{3t} x_2 = e^{-3t} Each gives a corresponding solution for y , using y = x' + 2x .
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