Question

(1 point) Math 216 Homework webHW3, Problem 11 Find the solution of the system where primes indicate derivatives with respect to t, that satisfies the initial condition x(0) - -2, y(0) - 5((-1/5)-(5/(10sqrt(30))))en(sqrt(30)t)-5(1/5)-(5/(10sqrt(C X- ysqrt(30)((-1/5)-(5/(10sqrt(30))e (sqrt(30)t)+sqrt(30)(1/ Based on the general solution from which you obtained your particular solution, complete the following two statements: The critical point (0,0) is A. unstable B. asymptotically stable C. stable and is a A. saddle point B. node ° C. Spiral D. center

Answer 1

Given x^t = 5y rightarrow (1) y^t = 6x rightarrow (2) (1) and (2) in operator form (D) x - 5y = 0 rightarrow (3) where D equiv d/dt (D) y - 6x = 0 rightarrow (4) Let us eliminate y from (3) and (4): pre multiplying eq^n (3) with (D) and multiplying eq^n (4) with (5) and then adding gives (D) (D) x - 30 x = 0 Rightarrow (D^2 - 30) x = 0 rightarrow (5), which is a linear differential equation with constant coefficients. The algebraic equation is m^2 - 30 = 0 Rightarrow m = plusminus squareroot 30 Complementary function of (5) is C.F = x_c = c_1 e^squareroot 30 t + c_2 e^- squareroot 30 t Since (5), is a homogeneous differential equation, general solution of (1) is x (t) = c_1 e^squareroot 30 t + c_2 e^- squareroot 30 t Substituting x in eq^n (1), we can greaterthanorequalto y. x^t = 5y squareroot 30 c_1 e^squareroot 30t - squareroot 30 c_2 e^- squareroot 30t = 5y Rightarrow