 # #1 Consider the nonlinear scalar differential equation x'(t) =ax(t...

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#1 Consider the nonlinear scalar differential equation x'(t) =ax(t)+f(t,x(t)),x(to =ro = where a is real and constant and the function f is continuous with f(t,0) = 0. Assume that there is a constant X such that If (t, x) < 1/x% Use the Lyapunov function V(x)=r2, = to show that a) the zero solution of (1) is stable provided that b) the zero solution of (1) is uniformly asymptotically stable (UAS) provided a + I< - for some positive constant a. c) Use part b) to show that the zero solution of + is (UAS). #2 Find a Lyapunov function to show the zero solution of = - - y' = - 2y + y (x2 + y2) is (U.A.S). #3 Find a Lyapunov function to show the zero solution of x = y = = - is stable. Hint: try V(x,y) = ax4 + by², and then chose a and b, accordingly. #4 Show the zero solution of r'=2ry+rd = y'=-r2+y = is unstable. #5 Consider =y-xf(x,y) y==r-yf(x,y) = - 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) = Were A(t) and B(t) are continuous on 0 prove that if all solutions are bounded, then they are stable.

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