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PAPER III: DIFFEENTIAL EQUATIONS

UNIT I
:

Systems of first order equations – existence and uniqueness theorem – Fundamental matrix - Non-homogeneous linear systems – linear systems with constant coefficients – linear systems with periodic co-efficients.

UNIT II:
Successive approximation – Picard’s theorem - Non-uniqueness of solution – Continuation and dependence on initial conditions, Existence of solutions in the large – Existence and uniqueness of solutions of systems.

UNIT III:
The Cauchy problem: The Cauchy problem – Cauchy – Kowlalewsky theorem – Homogeneous wave equation – Initial – Boundary value problems – Non-homogeneous boundary conditions – Finite string with fixed ends – Non-homogeneous wave equation.

UNIT IV:
Methods of separation of variables: Seperation of variables – The vibrating string problem – Existence and Uniqueness of solution of the vibrating string problem. The heat conduction problem – existence and uniqueness of solution of the heat conduction problem – The laplace and beam equations.

UNIT V:
Boundary value problems: Boundary value problems – Maximum and minimum principles – Uniqueness and continuity theorems – Dirichlet problems for a circle – Dirichlet problems for a circular annulus – Neumann problem for a circle Drirchlet problem for a rectangle – Dirichlet problem involving poission equation – Neumann problem for a rectangle.

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