Consider the system of differential equations dxdt 12x 075

Consider the system of differential equations dx/dt = -1.2x + 0.75y, dy/dt = 1.66666666666667x - 3.2y. For this system, the smaller eigenvalue is and the larger eigenvalue is . [Note-- you\'ll probably want to view the phase plotter at phase plotter (right click to open in a new window). Select the intergral curves utility from the main menu.] If y\' = Ay is a differential equation, how would the solution curves behave? The solution curves converge to different points. The solution curves would race towards zero and then veer away towards infinity. (Saddle) All of the solutions curve would converge towards 0. (Stable node) All of the solution curves would run away from 0. (Unstable node) The solution to the above differential equation with intial values x(0) = 4, y(0) = 7 is x(t) = . y(t) = .

Solution

given dx/dt = -1.2x +0.75y

integrate on both sides

x = -1.2x^2 / 2 + 0.75xy +c   (where c is an integration constant)

but given x(0) =4

x(0) = -0.6 (0)^2 + 0.75 (0) (4) + c

4 = 0+ 0 +c

so c=4

so x(t) = -0.6 x^2 +0.75xy +4

similerly dy/dt = 1.666667 x -3.2y

integrating on both sides

y(t) = 1.666667xy - 3.2 y^2 /2 + c

y(t) = 1.666667 xy - 0.6 y^2 +c

y(0) =7

7 = 0 -0 +c

so c=7

y(t) =1.666667 xy - 0.6 y^2 +c