Consider the fourth order Dirichlet (biharmonic) boundary value problem
y^(4) + [ lambda - q(t)] y = 0 in ( 0,1),
y(0) = y'(0) = 0
y(1) = y'(1) = 0
Where q : [0,1] -> R is continuous function. Prove that if phi(t, lambda 1) and phi(t, lambda 2) are solutions of this equation...
Consider the second order linear equation
z" + c(t)z = 0
Where c(t) is a continuous real-valued function of a real variable.
(a) Show that every (nontrivial) solution of this equation is non-oscillatory if c(t) < (1 - epsilon)/(4t^2) for t>=1, where epsilon > 0 is a real number...
Determine the asymptotic stability of the system x' = Ax where
A is 3 x 3 matrix
A = -1 1 1
0 0 1
0 0 -2
( first row is -1 1 1 second is 0 0 1 and third is 0 0 -2)
More specifically, what stability conclusion(s) can be drawn? ( Justify your answer)