Question

Series solution of differential equation

Answer 1

Answer:

Before we get into finding series solutions to differential equations we need to determine when we can find series solutions to differential equations. So, let’s start with the differential equation,

p(x)y″+q(x)y′+r(x)y=0

This time we really do mean nonconstant coefficients. To this point we’ve only dealt with constant coefficients. However, with series solutions we can now have nonconstant coefficient differential equations. Also, in order to make the problems a little nicer we will be dealing only with polynomial coefficients.

Now, we say that x=x_{0} is an ordinary point if provided both

\frac{{q\left( x \right)}}{{p\left( x \right)}}\,\,\hspace{0.25in}{\mbox{and}}\hspace{0.25in}\frac{{r\left( x \right)}}{{p\left( x \right)}}

are analytic at x=x_{0}. That is to say that these two quantities have Taylor series around x=x_{0}. We are going to be only dealing with coefficients that are polynomials so this will be equivalent to saying that

p\left( {{x_0}} \right) \ne 0

for most of the problems.

If a point is not an ordinary point we call it a singular point.

The basic idea to finding a series solution to a differential equation is to assume that we can write the solution as a power series in the form,

\begin{equation}y\left( x \right) = \sum\limits_{n = 0}^\infty {{a_n}{{\left( {x - {x_0}} \right)}^n}} \label{eq:eq2}\end{equation}

and then try to determine what the a_{n}’s need to be. We will only be able to do this if the point x=x_{0}, is an ordinary point. We will usually say that \eqref{eq:eq2} is a series solution around x=x_{0}.

Let’s start with a very basic example of this. In fact, it will be so basic that we will have constant coefficients. This will allow us to check that we get the correct solution.

Answer 2

$\huge\tt\orange{Answer:-}$

In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.....