## Transcribed Text

#1 Consider the nonlinear scalar differential equation x'(t)= ax(t) +f(t,x(t)),x(to)=x = , where a is real and constant and the function f is continuous with f(t,0) = 0. Assume that there is a constant > such that f (t.x)/<1/2). Use the Lyapunov function V(x==x, = to show that a) the zero solution of (1) is stable provided that a+à = 0. b) the zero solution of (1) is uniformly asymptotically stable (UAS) provided a + > < - for some positive constant a. c) Use part b) to show that the zero solution of = + 1 is (UAS). #2 Find a Lyapunov function to show the zero solution of x' = - x + y - x(x² + y2) y' = - 2y + y(x2 + y2) is (U.A.S). #3 Find a Lyapunov function to show the zero solution of x = y = Y==r3-y3 = - is stable. Hint: try V(x,y) = a.4 + by2 and then chose a and b, accordingly. #4 Show the zero solution of x' = 2xy + x³ y =-x²+y5 = - is unstable. #5 Consider = - where f is continuously differentiable. Show that a) if f(x,y) > 0, then the origin is A.S, b) whereas if f(x,y) < 0 the origin is unstable. Hint: Use appropriate polynomial Lyapunov function. #5 Consider the nonhomogenous system I' (t) = A(t):(t)+B(t), = Were A(t) and B(t) are continuous on 0 < to < t < 00, prove that if all solutions are bounded, then they are stable.

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