Question

4x - y, = 2x + y. Solve the system of differential equations with initial conditions x(0)=1, y(0)=2.

Answer 1

8. The ginen system of differential equations are de et = 4x - y e dy = 2x+y » (0-4) x + y = 0 ---- (1) - 2x + (0-1)y = 0 ---(2), where Da Multipling (1) by 2 and (2) by (0-4) and adding we get 24 + (0-1) (D-4) y = 0. (0²-50+6) y = 0 -- -- (3) xet y=emt be the trial solution of (3). Then auxiliary equation of 3) is m25m+6=0 → m-2) (n-3).= 0 m=2,3. : The complementary function of 3) is yo = Get+ est , where G A c, are - arbitrary constants : The general solution of (3) is y = Ye = 4 e 2t+G e 3t .. .: Dy = d = 29 e 2t + 3 est from (2) we get 2x = Dy - y 2x = 29e2+ + 36 e 3t_ geht gest * 3

2x = 9 e 2t + RG est a = a 2t + Geet The general solution of the system of differential equations are cot ni x(t) = q e 2t + G. est y(t) = G e 2 + + G e 3t, where a and a are arbitrary constants. using initial conditions x (o)=1& y(0) = 2 we have = a + 4 & 2 = GtG > 1= a + (2-6) > -1 = - 4 G=2 G = 0. The particular solution of the ginen system of differential equations with initial conditions is x (t) = ezt y(t) = 2e2t