Question

Question related to differential equations course....

Use method of un determinate coefficients

Answer 1

The method of undetermined coefficients is a method of finding a particular solution to certain non-homogeneous ordinary differential equations.

A guess is made as to the appropriate form, which is then tested by differentiating the resulting equation.

Consider a linear non-homogeneous ordinary differential equation of the form

where denotes the i-th derivative of y, and   denotes a function of X.

The method consists of finding the general homogeneous solution   for the complementary linear homogeneous differential equation

and a particular integral y(p)  of the linear non-homogeneous ordinary differential equation based on g(x). Then the general solution y to the linear non-homogeneous ordinary differential equation would be

If  g(x) consists of the sum of two functions h(x)+w(x) and we say that is the solution based on h(x) and the solution based on  w(x). Then, using a superposition principle, we can say that the particular integral is

EXAMPLE :

Find the general solution of the equation:

is a polynomial of degree 2, so we look for a solution using the same form,

substituting this particular function into the original equation yields,

which gives:

Solving for constants we get:

To solve for the general solution,

where is the homogeneous solution  , therefore, the general solution is: