Now let's return to our original problem of finding a particular solution of y sub p of homogeneous differential equation one, say P(d)y = g. We assume that g has an annihilator Q(D), Q(D) is a differential polynomial of degree m, okay, then we get the following theme, right? We are looking for a particular solution of P(D)y = g, okay, then I'm assuming that Q(D) g = 0, right? Then combining these two, apply Q(D) on both sides of this, then you get Q(D) P(D)y of p, okay? Which is equal to P(D)yp, that is a g, right, then Q(D)g = 0, okay? So we get this, what does that mean? That means yp is a solution to homogeneous problem Q(D)P(D)y = 0, okay? This is a homogeneous, linear, constant coefficient differential equation of order m + n, to remind you of that. This is the differential operator of order m, and this is a differential operator of order n, both have constant coefficients in the linear. So their product will be a differential polynomial of order n + m, right. Okay, and our particular solution that we are looking for is a solution to this homogeneous problem. So we are interested in general solution of Q(D)P(D)y = 0, we're interested in its general solution, okay? Finding a general solution of such a homogeneous differential equation is the same as finding its so-called fundamental set of solutions. The fundamental set of solutions of this differential equation is, okay, consists of m + n linearly independent solution, okay. We are looking for m + n, okay, linearly independent, Solutions to this problem, okay. We'd like to choose such the solutions in the following way. Choose the fundamental set of solutions of homogeneous differential equation Q(D) P(D)y = 0 in the form of y sub i union zj. Where i is moving from 1 to n, and j is moving from 1 to m, so together we have n + m linear independent solutions. In particular, I require then, the first set, y sub i, where i is moving from 1 to n, is a fundamental solution of P(D)y=0. If we choose those two sets, y sub i and g sub j, then the general solution of Q(D)P(D)y = 0 can be written as, y is equal to sum of c sub y, yi plus sum of dj times zj. Where the ci's and the dj's are arbitrarily constant, okay? So this is the function, this linear combination, equation two, okay? This represents a general solution of the Q(D)P(D)y = 0, okay? Using this fact with the following observation, okay? Any particular solution that they are looking for for the original problem, okay? Must satisfy the same homogeneous linear differential equation, Q(D)P(D)y = 0, okay? And we now know the general solution of such a differential equation given by the equation two. So what does that mean, right, yp must be of this form, for suitable ci and suitable dj, right? So let me write the claim here, okay, so our desired, the particular solution yp should be written as sum of i = 1 to n, c sub 1 y sub, plus sum of j = 1 to m, dj times gj, for some suitable coefficient c i's, and d j's right? Well, what does that mean, then? That means, if I apply the P(D) to the function y over p, you should get g, right? Because the y over p is the particular solution of that equation, okay? But using the expression above, yp is given by the linear combination. Sum of ci and yi plus sum of dj and times gj, right? What can you say about the action of a P(D) on the sum? Because I am assuming that all these yi's, they form a fundamental set of solutions of P(D)y = 0, okay? So, what does that mean [INAUDIBLE] P(D)? Acting on each yi that is equal to 0 for all i through 1 to the n, right? By the superposition principle, x over P(D) on this will be equal to 0. So that will be 0 plus P(D) and the sum of dj and the zj right. Where j is moving from 1 to m, right, so the particular solution we are looking for, right? Can we get one of such candidates here, okay? P(D) acting on some dj zj must be equal to g, for suitable choice of dj, that's what I mean. This is what I mean here, okay, so there must be a particular solution y sub P of P(D)y = g. Which is of the form yP that is equal to sum of j is equal to 1 to n, dj times g sub j, right. And where the coefficient dj will be determined by the condition saying P(D)yp, and that is equal to G, okay. We call this process the method of undetermined coefficient, right, okay? The suitable constant dj, the other and determine the coefficients which will be determined by equation g = p(D) acting on this linear combination, okay? I'll illustrate all these things, so there are several examples.
