Series solutions to differential equation, Mathematics

Assignment Help:

Before we find into finding series solutions to differential equations we require determining when we can get series solutions to differential equations. Therefore, let's start with the differential equation,

 p (x) y′′ + q (x) y′ + r (x )y = 0                   (1)

Now there we really do mean nonconstant coefficients. For this point we've only dealt along with constant coefficients. Though, with series solutions we can now contain nonconstant coefficient differential equations. As well as, in order to make the problems some nicer we will be dealing only along with polynomial coefficients.

Here, we say that x=x0 is an ordinary point if given both,

q(x)/p(x)                      and                  r(x)/p(x)

Both are analytic at x=x0. It is to say that such two quantities have Taylor series around x=x0. Our aim is only dealing with coefficients which are polynomials thus this will be equivalent to saying as,

p(x0) ≠ 0

If a point is not an ordinary point so we call this a singular point.

The fundamental idea to finding a series solution to a differential equation is to suppose that we can write the solution like a power series in the form,

1856_Series Solutions to Differential Equation9.png..................(2)

And then try to find out what the an's require to be. We will only be capable to do this if the point x=x0, is an ordinary point. We will generally say as (2) is a series solution around x=x0.

Let's begin with a very fundamental example of this. Actually this will be so fundamental that we will contain constant coefficients. It will permit us to check that we find the exact solution.


Related Discussions:- Series solutions to differential equation

Sketch several trajectories for the system, Sketch several trajectories for...

Sketch several trajectories for the system, x 1 ' = x 1 + 2x 2                                                                                x 2 ' = 3x 1 + 2x 2

Integration, what is integration and how is it important

what is integration and how is it important

What is transitive relations:, R is called as a transitive relation if (a, ...

R is called as a transitive relation if (a, b) € R, (b, c) € R → (a, c) € R In other terms if a belongs to b, b belongs to c, then a belongs to c.         Transitivity be uns

Determine the measure of the vertex angle, Determine the measure of the ver...

Determine the measure of the vertex angle of the isosceles triangle. a. 34° b. 16° c. 58° d. 112° d. Simply substitute x = 34 into the equation for the vertex angle,

Describe common phrases to represent math operations, Describe Common Phras...

Describe Common Phrases to Represent Math Operations? The table below shows the common phrases used in word problems to represent addition, subtraction, multiplication, and div

Innovation, In the innovations algorithm, show that for each n = 2, the inn...

In the innovations algorithm, show that for each n = 2, the innovation Xn - ˆXn is uncorrelated with X1, . . . , Xn-1. Conclude that Xn - ˆXn is uncorrelated with the innovations X

Most crucial aspect of learning multiplication, Which of the following is t...

Which of the following is the most crucial aspect of learning multiplication? i) Multiplication facts ii) Recall of tables and their recitation iii) Understanding "how man

Show that the angles subtended at the centre , A circle touches the sides o...

A circle touches the sides of a quadrilateral ABCD at P, Q, R and S respectively. Show that the angles subtended at the centre by a pair of opposite sides are supplementary.

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd