Course Objectives:
1. The course is designed to equip the students with the necessary mathematical skills and
techniques that are essential for an engineering course.
2. The skills derived from the course will help the student from a necessary base to
develop analytic and design concepts.
Course Outcomes: At the end of the Course, Student will be able to:
1. Solve linear differential equations of first, second and higher order.
2. Determine Laplace transform and inverse Laplace transform of various functions and use
Laplace transforms to determine general solution to linear ODE.
3. Calculate total derivative, Jocobian and minima of functions of two variables.
UNIT I:
Differential equations of first order and first degree:
Linear-Bernoulli-Exact-Reducible to exact.
Applications:
Newton’s Law of cooling-Law of natural growth and decay-Orthogonal trajectories- Electrical
circuits- Chemical reactions.
UNIT II:
Linear differential equations of higher order:
Non-homogeneous equations of higher order with constant coefficients with RHS term of the type eax, sin ax, cos
ax, polynomials in x, eax V(x), xV(x)- Method of Variation of parameters.
Applications: LCR circuit, Simple Harmonic motion.
UNIT III:
Laplace transforms:
Laplace transforms of standard functions-Shifting theorems - Transforms of derivatives and integrals – Unit step
function –Dirac’s delta function- Inverse Laplace transforms– Convolution theorem (with out proof).
Applications: Solving ordinary differential equations (initial value problems) using Laplace transforms.
UNIT IV:
Partial differentiation:
Introduction- Homogeneous function-Euler’s theorem-Total derivative-Chain rule-Generalized Mean value
theorem for single variable (without proof)-Taylor’s and Mc Laurent’s series expansion of functions of two
variables– Functional dependence- Jacobian.
Applications: Maxima and Minima of functions of two variables without constraints and Lagrange’s method
(with constraints).
UNIT V:
First order Partial differential equations:
Formation of partial differential equations by elimination of arbitrary constants and arbitrary functions –solutions
of first order linear (Lagrange) equation and nonlinear (standard types) equations.
UNIT VI:
Higher order Partial differential equations:
Solutions of Linear Partial differential equations with constant coefficients. RHS term of the type
byax m n
e ,sin(ax+by),cos(ax+by),x y
+
. Classification of second order partial differential equations.
1. The course is designed to equip the students with the necessary mathematical skills and
techniques that are essential for an engineering course.
2. The skills derived from the course will help the student from a necessary base to
develop analytic and design concepts.
Course Outcomes: At the end of the Course, Student will be able to:
1. Solve linear differential equations of first, second and higher order.
2. Determine Laplace transform and inverse Laplace transform of various functions and use
Laplace transforms to determine general solution to linear ODE.
3. Calculate total derivative, Jocobian and minima of functions of two variables.
UNIT I:
Differential equations of first order and first degree:
Linear-Bernoulli-Exact-Reducible to exact.
Applications:
Newton’s Law of cooling-Law of natural growth and decay-Orthogonal trajectories- Electrical
circuits- Chemical reactions.
UNIT II:
Linear differential equations of higher order:
Non-homogeneous equations of higher order with constant coefficients with RHS term of the type eax, sin ax, cos
ax, polynomials in x, eax V(x), xV(x)- Method of Variation of parameters.
Applications: LCR circuit, Simple Harmonic motion.
UNIT III:
Laplace transforms:
Laplace transforms of standard functions-Shifting theorems - Transforms of derivatives and integrals – Unit step
function –Dirac’s delta function- Inverse Laplace transforms– Convolution theorem (with out proof).
Applications: Solving ordinary differential equations (initial value problems) using Laplace transforms.
UNIT IV:
Partial differentiation:
Introduction- Homogeneous function-Euler’s theorem-Total derivative-Chain rule-Generalized Mean value
theorem for single variable (without proof)-Taylor’s and Mc Laurent’s series expansion of functions of two
variables– Functional dependence- Jacobian.
Applications: Maxima and Minima of functions of two variables without constraints and Lagrange’s method
(with constraints).
UNIT V:
First order Partial differential equations:
Formation of partial differential equations by elimination of arbitrary constants and arbitrary functions –solutions
of first order linear (Lagrange) equation and nonlinear (standard types) equations.
UNIT VI:
Higher order Partial differential equations:
Solutions of Linear Partial differential equations with constant coefficients. RHS term of the type
byax m n
e ,sin(ax+by),cos(ax+by),x y
+
. Classification of second order partial differential equations.