Reference no: EM132191680

1. On the interval (0,1], consider the differential equation ty' + y = q(t) where q ∈ C((0,1]) and lim_t→0 + q(t) = M. Show that the DE has a unique solution x*(t) satisfying the condition limsup_t→o+lx*(t)| < +∞.

2. Let u ∈ C(R₊; R₊) be a nonegative function satisfying the inequality

u(t) ≤ q(t) + ₀∫^tp(s)u(s) ds for t > 0,

where p, q ∈ C(R₊;R₊). Show that u admits an estimate

u(t) ≤ q(t) + ₀∫^texp (_s∫^t p(τ) dτ) p(s)u(s) ds for t > 0.

Furthermore, if q is a monotonically increasing function, u admits the estimate

u(t) ≤ q(t) exp (₀∫^t p(s) ds), fort > 0.

3. Show that the differential equation y' = 2 sin² t y + cos t has a unique periodic solution and find it.

4. Is the differential system: 

x' = -2x - y + z

y' = x - z

z' =  -x - y

5. Prove that no solution of the equation y' = t + y² can be extended on the infinite interval 0 < t < +∞.

6. Determine if the zero solution of the following scalar DE asymptotically stable/stable but not asymptotically stable or unstable:
(a) y' = y²
(b) y'' + 2ky' + y = 0, with k > O.
(c) What happens if either k < 0 or if k = 0 in problem (b).

7. Prove that if

A(t) ₀∫^t A(τ) dτ = (₀∫^t A(τ) dτ) A(τ) for t ∈ I

then

X (t) = exp (₀∫^t A(τ) dτ)

is a fundamental matrix of the system y' = A(t)y.

8. In the interval [1, ∞), consider y' = (sin(In t) + cos(ln t) - μ)y where 1 < µ < (1 - e(-Π/2))^-1. Show that the zero solution of this equation is asymptotically stable.