Our aim here is show how to solve an ordinary differential equation using power series and how to use techniques presented here to carry out further research/study based on project-based learning practices.
In this unit, we consider the possibility of representing solutions to linear differential equations in the form of some type of infinite series. In particular, differential equations whose solutions can be represented as a convergent power series, \begin{equation*} y(x) = \sum_{n=0}^{\infty} a_n x^n \end{equation*} where \( a_n \) are constants. This can be considered as a generalization of the method of undetermined coefficients for the case when we have an infinite number of constants. Our goal here is to determine the appropriate values of these constants by substitution into the differential equation.
There are three problem sets for this project:
Each problem set contains:
The student followed six steps of a project-based learning approach. These steps are:
There are twenty multiple choice questions in this problem set. Please click on this link.
There are twenty multiple choice questions in this problem set. Please click on this link.
Legendre polynomials first arose in the problem of expressing the Newtonian potential of a conservative force field in an infinite series involving the distance variable of two points and their included central angle. Other similar problems dealing with either gravitational potential or electrostatic potentials and steady-state heat conduction problems in spherical solids also lead to Legendre polynomials. Other polynomials which are commonly found in applications are the Hermite and Chebyshev. They play an important role in quantum mechanics, and in probability theory.
There are many second order equations occurring frequently in applied mathematics, engineering and physics, such as Bessel’s and Laguerre’s equations, etc. The solution solutions to these equations, that occur in applications, are referred to as special functions. They are called ‘special’ as they are different from the standard functions like sine, cosine, exponential, logarithmic, etc. The solution techniques presented here can be used in artificial intelligence and machine learning.