MTH303

MTH 303: Ordinary Differential Equations II 3 credits

1. Existence and uniqueness theory: Fundamental existence and uniqueness theorem. Dependence of solutions on initial conditions and equation parameters. Existence and uniqueness theorems for systems of equations and higher-order equations.

2. Series solutions of second order linear equations: Taylor series solutions about an ordinary point. Fobenius series solutions about regular singular points. Series solutions of Legendre, Bessel, Laguerre and Hermite differential equations, Hyper geometric equations.

1. Special functions: Gamma function. Error function. Hyper geometric function (Hyper geometric equation, special hyper geometric function, Generalized hyper geometric function, special confluent hyperbolic functions).

2. Legendre functions (Generating function, recurrence relations and other properties of Legendre polynomials, Expansion theorem, Legendre differential equation, Legendre function of first kind, Legendre function of second kind, associated Legendre functions).

3. Bessel functions (Generating function, recurrence relations, Bessel differential equation, Integral representations Orthogonality relations, Modified Bessel functions).

4. Hermite polynomials, Laguerre polynomials (Generating function, Rodrigue’s formula, orthogonal properties, Hermite and Laguerre differential equation, recurrence relations, expansion theorems). Laplace transforms: Basic definitions and properties, Existence theorem. Transforms of derivatives. Relations involving integrals. Transforms of Bessel functions.

7. Systems of linear first order differential equations: Elimination method. Matrix method for homogeneous linear systems with constant coefficients. Variation of parameters. Matrix exponential.

Evaluation: Incourse Assessment 30 Marks. Final examination (Theory, 3 hours). 70 Marks

Eight questions of equal value will be set, of which any five are to be answered.

References

1. S. L. Ross, Differential Equation

2. D. G. Zill, A First Course in Differential Equations with Applcations.

3. F. Brauer & J. A. Nohel, Differential Equations.

4. H.J.H. Piaggio, An Elementary Treatise on Differential Equations and Their Applications