## 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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