A function is periodic, with period t , if it repeats itself exactly after an interval of length t . i. e. y(x) = y(x+ Odes can be solved over an interval of length t subject to periodic boundary conditions, which state that the function and its derivatives at one end of the interval equal their values at the other end of the interval. The solution is then a periodic function with period t . For example, for a second-order ode to be solved over the interval 0 < x < t , two conditions are needed to uniquely specify the solution. The corresponding periodic boundary conditions are y(0) = y(t ) and y0(0) = y0(t ). e:g: y00 + y = (1 4 2) cos(2 x); The homogeneous and particular solutions are y(0) = y(1); y0(0) = y0(1) (so t = 1): yh = a cos x + b sin x & yp = cos(2 x):
