Differential Equations, Linear models

The boundary value problem ,Ty'' + p(w^(2))y' = 0 , y(0) = 0, y(L) = 0 is the model of the shape of a rotating string. Suppose T and ρ are constants. Find the critical angular speed w=W(n) for which there is non-trial solutions .

A force of 64 N stretches a spring 4 meters. The mass of 4 kg is attached to the spring and set into motion in a medium that offer a damped force equal 16 times the velocity .If the mass is pulled down 2 centimeters from the equilibrium position and released. Find the equation for the position....

Find the charge on the capacitor in an LRC series at t = 0.02 sec when L = 0.05 h, R = 2 Ohm, C = 0.01 farad. The initial conditions are q(0) = 5 C, i(0) = 0 Ampere. Determine the firt time at which the charge on the capacitor is equal to zero. Also, find the current at any time t.

Determine whether the given relation is an implicit solution to the given equation. Sin(y)+xy-x^3=2-----relation Y''=(6xy'+((y')^3)sin(y)-2(y')^2)/((3x^2)-y)-----equation The book says yes it is a solution but for some reason I am not getting it. You have to use implicit...

Just a quick question.