**Non Homogeneous Differential Equation:**

Now let us consider the following Non Homogeneous Differential Equation,

where the coefficients a_{0}, a_{1}, … a_{n} are constants, and *f*(t) is a function of me.

The general solution may be written

where x_{c} is the complementary function, and x_{p} is the particular integral. Since x_{c }is the general solution of the corresponding homogeneous equation with *f*(t) replaced by zero, we have to find out the particular integral x_{p}.

The particular integral can be calculated by the method of undetermined coefficients. This method is useful to equations

when c(t) is such that the form of a particular solution x_{p}Â of the above equation may be guessed.

For example, c(t) may be a single power of t, a polynomial, an exponential, a sinusoidal function, or a sum of such functions. The method consists in assuming for, x_{p} an expression similar to that of c(t), containing unknown coefficients which are to be determined by inserting x_{p} and its derivatives in the original equation.