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 U_{2}(x) if one linear independent solution U_{1}(x) is already known and the solution is to search for h(x) such that U_{2}(x)=U_{1}(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 U_{1}(x) and U_{2}(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** U_{p}(x) plus a linear combination of the solution of the homogeneous **differential equation**. Again the method of **variation of constants** was used to find U_{p}(x) of the form U_{p}(x)=U_{1}(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*.

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