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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)