Wednesday, 10 July 2013
Discussion Topic:
How many possible solutions of differential equation can be. Also discuss that how we can get a unique solution from an n-parameter family of solutions of a differential equation.
Answer:
The general solution to a differential equation must satisfy both the homogeneousand non-homogeneous equations. It is the nature of the homogeneous solution that the equation gives a zero value. If you find a particular solution to the non-homogeneous equation, you can add the homogeneous solution to that solution and it will still be a solution since its net result will be to add zero. This does not mean that the homogeneous solution adds no meaning to the picture; the homogeneous part of the solution for a physical situation helps in the understanding of the physical system. A solution can be formed as the sum of the homogeneous and non-homogeneous solutions, and it will have a number of arbitrary (undetermined) constants. Such a solution is called the general solution to the differential equation. For application to a physical problem, the constants must be determined by forcing the solution to fit physical boundary conditions. Once a general solution is formed and then forced to fit the physical boundary conditions, one can be confident that it is the unique solution to the problem, as gauranteed by the uniqueness theorem.
