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1. Power Series Solution of Differential Equations 
1. Power Series 
2. Ordinary and Singular Points
3. Power Series Solution about an Ordinary Point
4. Regular and Irregular in Singular Points
5. Power Series Solution about Regular Singular Point (Frobenius Method)
2. Bessel's Equations and Bessel's Funtions
1. Beta and Gamma Functions
2. Bessal's Differential Equation Bessel's Functions
3. Recurrence Relations for Bessel's Function
4. Generating Function for Bessel's Function
5. Orthogonality of Bessel's Functions
6. Integral Form of Bessel's Function
3. Legendre's Equation and Legendre's Functions
1. Legendre's Equation and Legendre's Function
1.1 Another Form of Pn (x)
2.Generating Function for Pn (x)
3. The Recurrence Relations for Pn (x)
4.Laplace's Definite Integral for Pn (x) 
5. Rodrigues' Formula
6.Orthogonal Property of Legendre's Polynimials
4. Partial Differential Equations of First Order 
1. Partial Differential Equations
2. Origin of First Order Partial Differential Equations
2.1 Formation of Partial Differential Equations by Elimination of Two Arbitary Function
2.2 Formation of Partial Differential Equation  by Elimination of Arbitary Function
3. Lagrange's Solution of The Linear Partial Differential Equations
4. Geometrical Interpretation of Lagrange's Linear Equations
5. Integral Surface Passing Through a given Curve 
6.Surfaces Orthogonal to given System of Surfaces
5.The Laplace Transforms
1. Definition of Extension of Laplace Transform 
2. Linearity Property
3. Laplace Transforms of SOme Elementary Functions
4. Shifting Properties 
5. Change of Scale Property
6. Laplace Transform of Derivatives
7. Multipilication by tn
8. Laplace Transform of Integrals
9. Division by t
10. Laplace Transform of Laplace Transform 
6. Inverse Laplace Transform 
1. Linear Property
2. Shifting Properties
3. Change of Scale Property
4. Inverse Laplace Transform of Derivatives
5. Division by s 
6. Inverse Laplace Transform of Integrals
7. Method of Partial Fractions to Evaluate Inverse Laplace Transform
8. The Convolution Property
7. Applications of Laplace Transforms 
1. Solution of Ordinary Differential Equations with Constant Coefficients
2. Solution of Ordinary Differential Equations with Variable Coefficients
3. Solution of Simultaneous Ordinary Differential Equations
DIFFERENTIAL EQUATIONS- II
Max. Marks : 30
Time : 3 Hours
Note: 1. The syllabus has been split into two Units: Unit-I and Unit-II. Four questions will be set
from each Unit.
2. A student will be asked to attempt five questions selecting at least two questions from each
Unit. Each question will carry 6 marks.
 
Unit-I
Series solution of differential equations-Power Series method, Bessel and Legendre equations.
Bessel functions of First and Second kind. Legendre function. Generating function. Recurrence
relation and orthogonality of Bessel and Legendre function.
Partial Differential Equations: Origin of first order Partial Differential Equations, Linear Equation of
first order, Integral surfaces passing through a given curve, surfaces orthogonal to a given system of surfaces.

Unit-II
Inverse Laplace transforms- Linearity property, Shifting properties, Change of Scale Property.
Inverse Laplace transforms of derivatives and integrals, Convolution theorem.
Applications of Laplace Transforms - Solution of differential equations with constant coefficients,
Solution of differential equations with variable coefficients, Solution of simultaneous differential equations.
Laplace Transformation-Linearity of the Laplace transformation. Existence theorem for Laplace
transformations, Shifting Theorems, Laplace transforms of derivatives and integrals, Multiplication of tn ,
Division by t .

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