Mathematics-III
Module-I (18 hours)
Partial differential equation of first order, Linear partial differential equation, Non-linear
partial differential equation, Homogenous and non-homogeneous partial differential
equation with constant co-efficient, Cauchy type, Monge’s method, Second order partial
differential equation The vibrating string, the wave equation and its solution, the heat
equation and its solution, Two dimensional wave equation and its solution, Laplace
equation in polar, cylindrical and spherical coordinates, potential.
Module-II (12 hours)
Complex Analysis:
Analytic function, Cauchy-Riemann equations, Laplace equation, conformal mapping,
Complex integration: Line integral in the complex plane, Cauchy’s integral theorem,
Cauchy’s integral formula, Derivatives of analytic functions
Module –III (10 hours)
Power Series, Taylor’s series, Laurent’s series, Singularities and zeros, Residue
integration method, evaluation of real integrals
Contents
Sl No Topics Page No
1.1 Formation of Partial Differential Equations
1.2 Linear partial differential equations of First Order
1.3 Non Linear P.D.Es of first order
1.3 Charpit’s Method
1.4 Homogenous partial differential Equations with constant coefficients
1.5 Non Homogenous partial differential Equations
1.6 Cauchy type Differential Equation
1.7 Monge's Method
2.1 One Dimensional wave equation
2.2 D Alemberts Solution of wave equation
2
,2.3 Heat Equation
2.4 Two Dimensional wave equation
2.5 Laplacian in polar coordinates
2.6 Circular Membrane ( Use of Fourier-Bessel Series)
2.7 Laplace’s Equation in cylindrical and Spherical coordinates
2.8 Solution of partial Differential equation by laplace Transform.
3.1 Analytic function
3.2 Cauchy –Reiman equation & Laplace equation
3.3 Conformal mapping
3.4 Line integral in complex plane
3.5 Cauchy’s integral theorem
3.6 Cauchy’s integral formula
3.7 Derivatives of analytic function
4.1 Power Series
4.2 Taylor series
4.3 Laurent Series
4.4 Singularities, Pole, and Residue
4.5 Residue Integral
4.6 Evaluation of real integral
Formation of Partial Differential Equations
In practice, there are two methods to form a partial differential equation.
(i) By the elimination of arbitrary constants.
(ii) By the elimination of arbitrary functions.
(i) Formation of Partial Differential Equations by the elimination of arbitrary
constants method:
* Let f(x,y,z,a,b)=0 be an equation containing 2 arbitrary constants ''a'' and ''b''.
3
, * To eliminate 2 constants we require at least 3 equations hence we partially
differentiate the above equation with respect to (w.r.t) ‘x’ and w.r.t ‘y’ to obtain 2
more equations.
* From the three equations we can eliminate the constants ''a'' and ''b''.
NOTE 1: If the number of arbitrary constants to be eliminated is equal to the
number independent variables, elimination of constants gives a first order partial
differential equation. But if the number of arbitrary constants to be eliminated is
greater than the number of independent variables, then elimination of constants
gives a second or higher order partial differential equation.
NOTE 2: In this chapter we use the following notations:
p = ∂z/∂x, q = ∂z/∂y, r = ∂2z/∂x2, s = ∂2z/(∂x∂y), t = ∂2z/∂y2
METHOD TO SOLVE PROBLEMS:
Step 1: Differentiate the given question first w.r.t ‘x’ and then w.r.t ‘y’.
Step 2: We know p = ∂z/∂x and q = ∂z/∂y.
Step 3: Now find out a and b values in terms of p and q.
Step 4: Substitute these values in the given equation.
Step 5: Hence the final equation is in terms of p and q and free of arbitrary constants ''a'' and ''b''
which is the required partial differential equation.
(ii) Formation of Partial Differential Equations by the elimination of arbitrary
functions method:
* Here it is the arbitrary function that gets eliminated instead of the arbitrary constants ''a'' and
''b''.
NOTE: The elimination of 1 arbitrary function from a given partial differential
equation gives a first order partial differential equation while the elimination of
the 2 arbitrary functions from a given relation gives second or higher order partial
differential equations.
4
, METHOD TO SOLVE PROBLEMS:
Step 1: Differentiate the given question first w.r.t ‘x’ and then w.r.t ‘y’.
Step 2: We know p = ∂z/∂x and q = ∂z/∂y.
Step 3: Now find out the value of the differentiated function (f'' ) from both the equations
separately. [(f’’) =?]
Step 4: Equate the other side of the differentiated function (f'' ) which is in terms of p in one
equation and q in other.
Step 5: Hence the final equation is in terms of p and q and free of the arbitrary function which is
the required p.d.e.
* Incase there are 2 arbitrary functions involved, then do single differentiation i.e. p = ∂z/∂x, q =
∂z/∂y, then also do double differentiation i.e.
r = ∂2z/∂x2, t = ∂2z/∂y2 and then eliminate
(f'' ) and (f'''' ) from these equations.
Worked out Examples
Elimination of arbitrary constants:
Ex 1:
5
Module-I (18 hours)
Partial differential equation of first order, Linear partial differential equation, Non-linear
partial differential equation, Homogenous and non-homogeneous partial differential
equation with constant co-efficient, Cauchy type, Monge’s method, Second order partial
differential equation The vibrating string, the wave equation and its solution, the heat
equation and its solution, Two dimensional wave equation and its solution, Laplace
equation in polar, cylindrical and spherical coordinates, potential.
Module-II (12 hours)
Complex Analysis:
Analytic function, Cauchy-Riemann equations, Laplace equation, conformal mapping,
Complex integration: Line integral in the complex plane, Cauchy’s integral theorem,
Cauchy’s integral formula, Derivatives of analytic functions
Module –III (10 hours)
Power Series, Taylor’s series, Laurent’s series, Singularities and zeros, Residue
integration method, evaluation of real integrals
Contents
Sl No Topics Page No
1.1 Formation of Partial Differential Equations
1.2 Linear partial differential equations of First Order
1.3 Non Linear P.D.Es of first order
1.3 Charpit’s Method
1.4 Homogenous partial differential Equations with constant coefficients
1.5 Non Homogenous partial differential Equations
1.6 Cauchy type Differential Equation
1.7 Monge's Method
2.1 One Dimensional wave equation
2.2 D Alemberts Solution of wave equation
2
,2.3 Heat Equation
2.4 Two Dimensional wave equation
2.5 Laplacian in polar coordinates
2.6 Circular Membrane ( Use of Fourier-Bessel Series)
2.7 Laplace’s Equation in cylindrical and Spherical coordinates
2.8 Solution of partial Differential equation by laplace Transform.
3.1 Analytic function
3.2 Cauchy –Reiman equation & Laplace equation
3.3 Conformal mapping
3.4 Line integral in complex plane
3.5 Cauchy’s integral theorem
3.6 Cauchy’s integral formula
3.7 Derivatives of analytic function
4.1 Power Series
4.2 Taylor series
4.3 Laurent Series
4.4 Singularities, Pole, and Residue
4.5 Residue Integral
4.6 Evaluation of real integral
Formation of Partial Differential Equations
In practice, there are two methods to form a partial differential equation.
(i) By the elimination of arbitrary constants.
(ii) By the elimination of arbitrary functions.
(i) Formation of Partial Differential Equations by the elimination of arbitrary
constants method:
* Let f(x,y,z,a,b)=0 be an equation containing 2 arbitrary constants ''a'' and ''b''.
3
, * To eliminate 2 constants we require at least 3 equations hence we partially
differentiate the above equation with respect to (w.r.t) ‘x’ and w.r.t ‘y’ to obtain 2
more equations.
* From the three equations we can eliminate the constants ''a'' and ''b''.
NOTE 1: If the number of arbitrary constants to be eliminated is equal to the
number independent variables, elimination of constants gives a first order partial
differential equation. But if the number of arbitrary constants to be eliminated is
greater than the number of independent variables, then elimination of constants
gives a second or higher order partial differential equation.
NOTE 2: In this chapter we use the following notations:
p = ∂z/∂x, q = ∂z/∂y, r = ∂2z/∂x2, s = ∂2z/(∂x∂y), t = ∂2z/∂y2
METHOD TO SOLVE PROBLEMS:
Step 1: Differentiate the given question first w.r.t ‘x’ and then w.r.t ‘y’.
Step 2: We know p = ∂z/∂x and q = ∂z/∂y.
Step 3: Now find out a and b values in terms of p and q.
Step 4: Substitute these values in the given equation.
Step 5: Hence the final equation is in terms of p and q and free of arbitrary constants ''a'' and ''b''
which is the required partial differential equation.
(ii) Formation of Partial Differential Equations by the elimination of arbitrary
functions method:
* Here it is the arbitrary function that gets eliminated instead of the arbitrary constants ''a'' and
''b''.
NOTE: The elimination of 1 arbitrary function from a given partial differential
equation gives a first order partial differential equation while the elimination of
the 2 arbitrary functions from a given relation gives second or higher order partial
differential equations.
4
, METHOD TO SOLVE PROBLEMS:
Step 1: Differentiate the given question first w.r.t ‘x’ and then w.r.t ‘y’.
Step 2: We know p = ∂z/∂x and q = ∂z/∂y.
Step 3: Now find out the value of the differentiated function (f'' ) from both the equations
separately. [(f’’) =?]
Step 4: Equate the other side of the differentiated function (f'' ) which is in terms of p in one
equation and q in other.
Step 5: Hence the final equation is in terms of p and q and free of the arbitrary function which is
the required p.d.e.
* Incase there are 2 arbitrary functions involved, then do single differentiation i.e. p = ∂z/∂x, q =
∂z/∂y, then also do double differentiation i.e.
r = ∂2z/∂x2, t = ∂2z/∂y2 and then eliminate
(f'' ) and (f'''' ) from these equations.
Worked out Examples
Elimination of arbitrary constants:
Ex 1:
5
