Mathematical methods Lecture 24 of 34

November 6, 2012 by Multimedia Publications and Printing Services

Lecturer: K.S. Narain , ICTP

This lesson continues with the study of differential equations, in particular with the second order inhomogeneos differential equations, it was shown that to solve the inhomogeneous differential equation is always useful to solve first the problem for the homogeneous differential equation. In order to find the general solution to this problem a determinant called the Wronskian was introduced. It was demonstrated that the 2nd order linear differential equation can only have two linear independent solutions; all the other possible solutions can be written as a linear combination of this two solutions.

Then the method called the variation of constants was introduced. This method allowed to obtain  the second possible linear independent solution U2(x) if one linear independent solution U1(x) is already known and the solution is to search for h(x) such that U2(x)=U1(x)h(x). In the fallowing a general expression for h(x) was derived and the Wronskian for this two solutions calculated and if was demonstrated that as the Wronskian is different from zero U1(x) and U2(x) are linear independent. In the case of the inhomogeneous differential equation it was shown that the general solution is a particular solution of the inhomogeneous differential equation Up(x) plus a linear combination of the solution of the homogeneous differential equation. Again the method of variation of constants was used to find Up(x) of the form Up(x)=U1(x)V(x)  and a general solution for V(x) was also obtained.

The second part of the lesson was dedicated to the introduction of Green's functions a  function used to solve inhomogeneous differential equations subject to specific boundary conditions. To explain what a Green's functions is some concepts were introduced like the differential operator and the adjoint of an operator. It was shown that if L is a linear differential operator then by definition <v|L|u>= <u|L+|v> where L+ is the adjoin operator of L.

Mathematics for Physicists (Text Book from Google e-books preview)

Written by Philippe Dennery,Andre Krzywicki, Chapter 1, 2, 3 and 4

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