Question

What is the technique of variation of parameters doing? What does it assume about the solution? How does it compare with undetermined coefficients? (What do they calculate? Do they give you the same thing? How are they different?)

Answer 1

The method of Variation of Parameters and the method of Undermined Coefficients are general methods to compute particular solution to nonhomogeneous linear ordinary as well as partial differential equations. The method of Variation of Parameters depends on integration, whereas the method of Undermined Coefficients is purely algebraic.

The method of Variation of Parameters has no prior conditions to be satisfied and thus is more general method compared to the method of Undermined Coefficients, which will work only for a small class of functions. The disadvantages of the method of Variation of Parameters are as follows:
(i) For the method of Variation of Parameters, the Complementary Function is required, whereas for the method of Undermined Coefficients, the Complementary Function is not required,

(ii) For the method of Variation of Parameters, while it is possible to write down a formula to get a particular solution, we may not be able to actually find it if the the integrals are too difficult.