String Standing Waves and Oscillator Solutions Without Newton's Second Law?

Newton’s second law or alternatively conservation of energy may be used in general to establish an equation of motion for a particle with rest mass in a nonrelativistic situation. In two special cases linked to periodic motion, namely an oscillating string tied at both ends and an oscillator, we try to argue that the dynamics of the problem follow from an orthogonality dot product (i.e. integral) of a function y(x) for the string and x(t) for the oscillator and its change i.e. dy/dx or dx/dt. This is equivalent to y and its change dy/dx (or x and dx/dt) forming the sides of a right angle triangle. This seems to be enough to solve these problems and suggests that only a first derivative is needed in such cases