Advanced Engineering Mathematics

Name: Advanced Engineering Mathematics
Rating: 3.8 (13 reviews)

For Engineers and Hobbyists

Created byDr. Rauf Tailony

Last updated 10/2019

English

What you'll learn

The objective of this course is to help you build the skill necessary to analyze mathematical relationships, and equations they encounter in the real world. Topics include Ordinary Differential Equations, Laplace Transforms, Systems of Linear Differential Equations

Course content

3 sections • 24 lectures • 3h 52m total length

Introduction to Linear differential equation11:35
Learn to solve first-order linear differential equations using the integrating factor method, recognizing equations of the form y' + p(x) y = q(x) and applying g(x) = e^{∫ p(x) dx}.
Solving linear 1st order differential equation using integrating factor9:56
Solve a linear first order differential equation using the integrating factor, identify p(x) and q(x), and derive y = (1/3) e^{3x} + C e^{x} from y(0) = -3.
Separable differential equations7:09
Homogeneous differential equations4:59
Bernoulli equation13:35
Identify the Bernoulli equation in the form y' + p(x) y = r(x) y^alpha, with alpha as a constant, and transform to a linear equation using new = y^(1-alpha).
Riccati equation8:21
Solve Riccati equations by using a known particular solution and the substitution y = s(x) + 1/z, turning the problem into a linear ODE, as shown in the example.
Exact equations- part 1- 1st order homogeneous differential equations13:32
Exact equations- part 2- 1st order homogeneous differential equations10:44
The lecture shows solving a non separable differential equation by using an integrating factor to make it exact, leading to the solution phi = x^2 y^3 - 2 x^2 + C.

Reduction of order- 2nd order homogeneous differential equations9:28
Apply reduction of order to a homogeneous second order differential equation by deriving y2 from y1 and forming y with C1 y1 + C2 y2; example y2 = x^2 ln|x|.
Exponential guess- Repeated roots- 2nd order homogeneous differential equations4:01
Exponential guess- Distinct roots- 2nd order homogeneous differential equations4:43
Apply the exponential guess method to a second order homogeneous differential equation with distinct roots, solving the example y''-y'-6y=0 to yield y = C1 e^{3x} + C2 e^{-2x}.
Exponential guess- complex roots- 2nd order homogeneous differential equations7:03
Master solving second order homogeneous differential equations with the exponential guess method, using the discriminant a^2−4b to determine complex roots and y = e^{αx}(C1 cos βx + C2 sin βx).
Euler - Cauchy equation7:27
Learn to solve the Euler-Cauchy equation by using y = x^m, which yields m^2 + (a-1)m + b = 0 and y = C1 x^{m1} + C2 x^{m2}.
Euler - Cauchy equation - repeated and complex roots9:13
Variation of parameters -2nd order NON-homogeneous differential equations14:21
Explore variation of parameters for a non-homogeneous differential equation, solve the homogeneous part, build the particular solution with u(x) and v(x) using the wronskian, and work through an example y''+4y=2x.
Undetermined coefficients-2nd order NON-homogeneous differential equations14:22
Learn to solve non-homogeneous differential equations with undetermined coefficients, building the homogeneous and particular solutions from a table of forms. See the y''-6y'+9y=5e^{3x} example.
Super position method-2nd order NON-homogeneous differential equations12:28
Learn to apply the superposition method to 2nd order non-homogeneous differential equations by splitting the right-hand side into f1 and f2, solving each, then combining homogeneous and particular solutions.

Introduction to the Laplace transform7:12
Explore the basics of the Laplace transform, its integral form, and how a transform table shows e^{a t} equals 1/(s−a). Apply the inverse transform to recover f(t).
Initial value problem - Laplace transform11:10
Learn to solve initial value problems with the Laplace transform by transforming derivatives, applying the transform rules, and using inverse Laplace to obtain y(t).
Using partial fraction expansion in the Laplace transform14:59
Repeated roots- partial fraction expansion- Laplace transform15:14
Explore repeated roots in Laplace transforms via partial fraction expansion and solve y''-4y'+4y=cos t, yielding y(t) = (22/25)e^{2t} - (13/15)t e^{2t} + (3/25)cos t - (4/25)sin t.
The convolution theorem5:39
The convolution theorem states that the Laplace transform of a convolution equals the product of transforms, and the inverse yields the convolution; an example uses f(t)=1 and g(t)= t e^{4t}.
Impulse and delta function transformation7:46
Explore the impulse and delta function transformation, defining the Dirac delta and its Laplace transform. Apply shifting properties to solve a second-order ode with delta(t-3) input.
Solving system of equations7:41
Apply the Laplace transform to solve systems of equations with x, x', y, y', including homogeneous and non-homogeneous terms, using partial fractions to obtain x(t) and y(t) from initial conditions.

Requirements

Calculus

Description

The objective of this course is to help you build the skill necessary to analyze mathematical relationships, and equations you encounter in the real world. Topics include Ordinary Differential Equations, Laplace Transforms, Systems of Linear Differential Equations. Upon completion of this course you should be able to identify different types of differential equations and decide the best solution method to follow in order to solve that equation. Differential equations are in most cases the mathematical representation of a real world problem in physics and engineering. Learning how to solve a differential equation or a Laplace transformation could be a solution to a problem that is facing the real world today.

Who this course is for:

Engineers, Applied engineering employees, Engineering students and hobbyists, Mathematics student, Physics students and hobbyists.

Advanced Engineering Mathematics

What you'll learn

Explore related topics

Course content

First order differential equations8 lectures • 1hr 20min

Second order differential equations9 lectures • 1hr 23min

The Laplace Transform7 lectures • 1hr 10min

Requirements

Description

Who this course is for: