Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 6 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Solutions Solution 1. When fl Y 0 the equations are dx dy dt rxy dt -rxy. Adding these two equations we see that dx dy dt dt Integrating and applying the initial conditions x 0 xo and y 0 yo we obtain x xo yo - y Substituting this into the differential equation for y d dt dy dt -r xo yo - y y -r xo yo y ry2. 2174 We recognize this as a Bernoulli equation and make the substitution u y 1. -y2ddy r x y y1 - r du 7 X r x y u - r dt d e-r x0 y0 tu -re-r x0 y0 t dt v 7 u er x0 y0 t -re-r x0 y0 t dt cer x0 y0 t u -------- cer x0 y0 t x y 1 -1 y cer x0 y0 t x y J Applying the initial condition for y 1 A-1 . c y x y J c ---------1--- y x y The solution for y is then y 1 I 1 I er x0 y0 t x y y x y -1 Since x x y - y the solution to the system of differential equations is x x y - Ấ I 1 1 - er x0 y0 t y x y 7 -1 y Ấ I 1 1 - er x0 y0 t y x y 7 -1 .
