We have solved canonical equation (1) by method of successive approximations by Pickar and obtained two fundamental solutions in the form of sequences - iterations.
When also diverges. Solutions (2) are plain Euclid's sine and cosine for a(x)=1. For the positive a(x), they are sufficiently high to cause oscillations, being some non elementary functions close to Euclid's sine and cosine, which we have called sine and cosine with base a(x) or generated sine and cosine. By applying theorem on mean value of integrals, method of geometric means and method of small interval, we have determined very precise approximate formulae for oscillating solutions of the equation (1) of the following type.
The approximate solutions (3) provide rather elementary operational formulae for addition theorems, derivatives, integrals etc. for which the function has the crucial role and they can also be extremely helpful for determination of oscillating solutions of differential equations of the following form: |