Consider the Laguerre differential equation x2y′′+x(1−x)y′+Nxy=0, (11.5.40) where N is a constant. Show that in the case when N is a positive integer, Equation (11.5.40) has a solution that is a polynomial of degree N, and find it. When properly normalized, these solutions are called the Laguerre polynomials.

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.