Lecture-wise plan of “Differential Equations”

A: Course Description
Topics Covered:
Classifications of Differential Equations, First Order Linear Equations: Linear Equations,
Separable Equations, Exact Equations and Integrating Factors, Homogeneous Equations,
Applications, Second Order Linear Equations: Methods for Solving Homogeneous Equations
and Non-homogeneous Equations, Linear Independence and the Wronskian. Applications,
Series Solutions: Review of Power Series, Series Solutions near an Ordinary Point, Regular
Singular Points, Euler Equations, Series Solutions near a Regular Singular Point, Laplace
Transform: Solution of Initial Value Problems, Partial Differential Equations: Variable
Separable Method, The Heat equation. The Wave Equation.

Course Objectives:
The course is designed to teach Differential Equations and Linear Equations, their practical
uses and applications. It will help the students to apply elementary solution techniques and be
able to interpret the results and solve specific problems in major area of studies
Students Learning Outcomes:
By the end of the course students will be able to:
1. Model a simple physical system to obtain a first order differential equation.
2. Test the plausibility of a solution to a differential equation (DE) which models a
physical situation by using reality-check methods such as physical reasoning, looking at
the graph of the solution, testing extreme cases, and checking units.
3. Find and classify the critical points of a first order autonomous equation and use them to
describe the qualitative behaviour and, in particular, the stability of the solutions.
The main equations studied in the course are driven first and second order constant
coefficient linear ordinary differential equations and 2x2 systems. For these equations
students will be able to:
4. Use known DE types to model and understand situations involving exponential growth
or decay and second order physical systems such as driven spring-mass systems or LRC
circuits.
5. Solve the main equations with various input functions including zero, constants,
exponentials, sinusoids, step functions, impulses, and superpositions of these functions.
6. Understand and use fluently the following features of the linear system response:
solution, stability, transient, steady-state, amplitude response, phase response,
amplitude-phase form, weight and transfer functions, pole diagrams, resonance and
practical resonance, fundamental matrix.
7. Use the following techniques to solve the differential equations described above:
characteristic equation, exponential response formula, Laplace transform, convolution
integrals, Fourier series, complex arithmetic, variation of parameters, elimination and
anti-elimination, matrix eigenvalue method.
8. Understand the basic notions of linearity, superposition, and existence and uniqueness
of solutions to DE's, and use these concepts in solving linear DE's.

Text Book:
A First Course of Differential Equations by D. G. Zill (9th Edition).

Reference Book(s):
1. Differential equations with boundary value problems By D. G Zill and M. R. Cullen,
Seventh Edition
2. Differential Equation: An Introduction to modern methods and Applications By Brannan
J., Boyce W.,
B: Lecture-wise Activities
Week # Lecture # Topic (Chapter/Topics Covered) Assignment Remarks

1 1 Introduction to Differential Example:

Equation Students are
 Definition of Differential advised to
Equation review basics
 Order of Differential ● Definition of
Equation the derivative
● Rules of
 Degree of Differential
differentiation
Equation.
● Derivative as
 Classification of a rate of
Differential Equations change
by types, order and ● First
linearity (Linear and derivative and
nonlinear, increasing/decr
homogeneous and easing
nonhomogeneous, ● Second
ordinary and patrial D. derivative and
E.) concavity)
 The normal form of
D.E.
2  Initial conditions
 Boundary
(Date) conditions
 Initial value
problem(IVP)
 Boundary value
problem
 The interval of
existence, or the
interval of validity
 Verification of
solution
 Solution curve
 Function versus
Solution
 Explicit and
implicit solutions
 Families of solution
 An IVP Can Have
Several Solutions
2 3  Autonomous first-order D.E. Quiz #1
 Introduction of
(Date) Separable differential
Equations
 Method to solve the
separable D.E.
 Examples
 D. E. as Mathematical
models
 Applications(Population
Growth and Decay,
Newton’s Law of cooling)
4  Definition of first order
linear Differential
(Date) equations
 Homogeneous,
nonhomogeneous linear
DE
 Standard form
 Property (The sum of two
solutions)
 Particular and complementary
solution
 Integrating Factor
 Steps for solving a linear first
order DE
 Constant of integration
 General solution
 Discontinuous co-
efficient
3 5  Differential of a
function of two
(Date) variables
 Definition of exact
differential
equations
 Criterion for an exact
differential equation
 Method of solution of
exact differential
equation
 Non-Exact DE
 Integrating Factors to
make the exact DE
 Examples
6  Definition of
homogeneous
(Date) differential
equations
 Substitutions for
solving homogeneous
DE
 Method for solving
homogeneous DE
 Examples
 Applications
(Bernoulli’s DE)
4 7  Definition of second
Order Linear Equations
(Date)  Method for solving second
order Linear DE
 Examples
 8  Existence of unique
solution
 A BVP have many, one, or no
solutions
 Differential Operator
 Anhilator operator
 nth-order differential operator
or polynomial operator
 Superposition principle
 Linearly independence
 Wronskian
 Fundamental Set of Solutions
 Method for Solving
Homogeneous Equations
 Examples and Applications
5 9  General solution Quiz #2
for non-
(Date) homogeneous DE
 Complimentary
functions
 Another superposition
principle
 Method for Solving
Non-Homogeneous
Equations
 Examples and
Applications
10  Reduction of order
method (general case
(Date) and standard form)
 A Second Solution by
Reduction of Order
 Method for solving
homogeneous DE
 Examples
6 11
 Review of above concepts
(Date)
12 Ist
Sessional
(Date)
7 13 Homogeneous linear DE
with constant coefficients
(Date)  Auxiliary equation
 Three cases of
auxiliary equation
(distinct, repeated
and complex
roots)
 Euler formula
 Examples

14 Homogeneous linear DE
with constant coefficients
(Date)  Higher orders DE
 Combination of
three cases
 Examples
 Applications

8 15 Method of Undetermined
coefficients
(Date)  General Solution Using
Undetermined
Coefficients
 Particular Solution
Using Undetermined
Coefficients
 Trial particular
solutions
 Different cases for
particular solutions by
superposition
 Examples

16 Method of Undetermined
coefficients
 Some cases for articular
solution
 Using the multiplication
rule
Examples
9 17  Review of power series Quiz 3
 Convergence
(Date)
 Interval of convergence
 Identity property
 Ordinary and singular
points
 Power series Solutions
 Examples
18  Review of Power Series
 Recurrence Relations
(Date)
 Examples
10 19  Series Solutions near an
Ordinary Point
(Date)  Three times recurrence
relation
 Examples
20  Regular Singular Points
 Non-polynomial
(Date) coefficients
 Examples
11 21 Second
sessional
(Date)
22 Euler Equations
Examples
(Date) Series Solutions near a Regular
Singular Point
Examples
12 23 Laplace Transform
 Definition
(Date)  Linearity property
 Integral transform
 Transforms of some
basic functions
 Examples
24 Laplace Transform
 Laplace is a linear
(Date) Transformation
 Sufficient condition for
existence
 Transformation of
piecewise continuous
function
 Examples
13 25 Inverse Laplace Transform and
Transforms of derivative
(Date)  Definition
 Some Inverse
transforms
 Basic theorem
 Partial fraction and
Inverse transformation
26 Partial Differential Equations.
 Definition
(Date)  Types of PDE
Parabolic
Elliptic
Hyperbolic
14 27 Partial Differential Equations. Quiz #4
Method for solution of
(Date) PDE
28  Variable Separable Method
 Examples
(Date)
15 29
The Heat equation
(Date)
30  The Heat equation
(Date)
16 31 The Wave Equation

(Date)
32  Review

(Date)
17 Final
(Date)