I think it's just because we don't want to look for particular solution that has a complicated form. The proof shows that when you apply certain conditions, the form of the particular solution is valid in most (if not all) cases.
Hello. I'm having a bit of trouble understanding the rationale behind the assumption that is made in the variation of parameters method for first and second order linear differential equations.
For the first order equations, my book lays out the property of linear equations that the solution is the sum of the complementary and a particular solution, that is . Then the author shows that the homogeneous equation
is seperable and has as its solution and letting for convenience, we have .
Now the author states that in order to find the particular solution we use a procedure known as variation of parameters where we assume that .
So, I have no trouble at all with the derivation from here on. My question is, how is it that this assumption is made. Or, stated in another way, what was the reasoning behind making this assumption. What is it that would make one believe that the particular solution would be simply some function of the independent variable times the complementary solution?
multiplied by the homogeneous ODE which is always equal to 0. The hope then is that this new ODE for would be solvable.
So as in your example suppose solve the homogeneous system
and you want to solve
Using this assumption as you get
now putting this into the ode gives