Question

what is meant by series solution of differential equation? why do we need series solution?

Answer 1

A series solution to a differential equation is one of the form\[y=\sum_{n=0}^{\infty}a_nx^n\]assuming you have some differential equation where you're looking to solve for y(x). You then substitute the series into the differential equation with the aim of finding a recurrence relation for the coefficients.

Series solutions are extremely useful since not all differential equations can be solved easily, or exactly, for some function...but the worlds of mathematics, science, engineering and economics all throw up differential equations that need to be solved quickly for use - this is where series solutions come into their own; when you don't know how to solve an equation, or have the time for insight, plug in a series solution and see what happens.

It must be stipulated, though, that certain assumptions need to be met before applying series.