Differential Equations Answers

Solve the simultaneous differential equations dx/dt + ky = 0 and dy/dt + kx = 0 with the initial conditions that x = 0 and y = 1 when t = 0. Hence, show that for all values of t, y^2 - x^2 = m where m is a constant to be determined.

If p = dy/dx, show that d^2y/ dx^2 = p(dp/dy).
Hence find the solution y = f(x) of the differential equation y(d^2y/dx^2) = 2(dy/dx) + (dy/dx)^2.

A particle A moves in a resisting medium in a straight line such that its distance x from a fixed point O satisfies the equation d^2x/dt^2 + p(dx/dt) + qx = 0, where p and q are constants. Find the condition(s) on p and q such that the motion of A is
(i) simple harmonic.
(ii) damped harmonic.
In the case where the motion is damped harmonic, find
(iii) the damping factor.
(iv) the period of the motion.