Question

I am desperate to know How to find or guess Particular solution(Yp) of Non-Homogeneous higher order differential equations,

Please provide me with explanations.

There is Key Identities method also to find Yp. But I can't understand.

Answer 1

To find a particular solution to a nonhomogeneous differential equation ;

y′′+p(t)y′+q(t)y=g(t)

One of the main advantages of this method is that it reduces the problem down to an algebra problem. The algebra can get messy on occasion, but for most of the problems it will not be terribly difficult. Another nice thing about this method is that the complementary solution will not be explicitly required, although as we will see knowledge of the complementary solution will be needed in some cases and so we’ll generally find that as well.

There are two disadvantages to this method. First, it will only work for a fairly small class of g(t)’s. The class of g(t)’s for which the method works, does include some of the more common functions, however, there are many functions out there for which undetermined coefficients simply won’t work. Second, it is generally only useful for constant coefficient differential equations.

The method is quite simple. All that we need to do is look at g(t) and make a guess as to the form of YP(t) leaving the coefficient(s) undetermined (and hence the name of the method). Plug the guess into the differential equation and see if we can determine values of the coefficients. If we can determine values for the coefficients then we guessed correctly, if we can’t find values for the coefficients then we guessed incorrectly.