Explore a comprehensive 12-hour introductory course on differential equations. Begin with an introduction to separable variables, followed by extensive practice solving example problems. Progress to linear equations, exact equations, and the Bernoulli equation, with real-world applications. Dive into advanced topics such as reduction of order, homogeneous and non-homogeneous linear differential equations with constant coefficients, and the Cauchy-Euler equation. Learn various problem-solving techniques, including the annihilator approach, variation of parameters, and substitution methods. Study systems of linear differential equations, nonlinear DEs, and modeling concepts. Explore free undamped and damped motion, driven motion with damping, and resonance. Conclude with an introduction to power series, the theorem of Frobenius, and Laplace transforms, providing a solid foundation in differential equations and their applications.
Overview
Syllabus
A1 Introduction.
B01 An introduction to separable variables.
B02 Example problem with separable variables.
B03 Example problem with separable variables.
B04 Example problem with separable variables.
B05 Example problem with separable variables.
B06 Example problem with separable variables.
B07 Example problem with separable variables.
B08 Introduction to linear equations.
B09 Example problem with a linear equation.
B10 Example problem with a linear equation.
B11 Example problem with a linear equation.
B12 Example problem with a linear equation.
B13 Example problem with a linear equation.
B15 Example problem with a linear equation using the error function.
2_2_8 Example problem with a linear equation modeling a real world situation.
B16 Introduction to exact equations.
B17 Example problem solving an exact DE.
B18 Example problem solving an exact equation.
B19 Example problem solving for an exact equation.
B20 Creating an exact equation by using an integrating factor.
B21 Example problem where an exact DE has to be created.
B22 Introduction to Substitutions.
B23 Example problem solving for a homogeneous DE.
B24 Introduction to the Bernoulli Equation.
B25 Example problem solving for a Bernoulli equation.
B26 U substitution.
B27 Introduction to linear models.
B28 An example problem of a linear model.
C01 Preliminaries.
C02 Reduction of order.
C03 Example problem using reduction of order.
C04 Example problem using reduction of order.
C05 Example problem using reduction of order.
C06 Example problem using reduction of order.
C07 Homogeneous linear differential equations with constant coefficients.
C08 Homogeneous linear differential equations with constant coefficients.
C09 Example problem solving a second order LDE with constant coefficients.
C10 Example problem solving a second order LDE with constant coefficients.
C11 Example problem solving a second order LDE with constant coefficients.
C12 Example problem solving a second order LDE with constant coefficients.
C13 Third and higher order linear DE with constant coefficients.
C14 Example problem with a third order linear DE with constant coefficients.
C15 Initial value problem solving a homogeneous linear SOODE with constant coefficients.
C16 Example of solving a homogeneous higher order linear ODE with constant coefficients.
C17 Non homogeneous higher order linear ODEs with constant coefficients.
C18 Example problem using the superposition approach.
C19 Example problem using the superposition principle.
C20 Example problem using the superposition principle.
C21 The annihilator approach.
C22 Simple examples finding the annihilator.
C23 More about the annihilator approach.
C24 Finding the differential annihilator.
C25 Solving a DE with the annihilator approach.
C26 Example problem finding the form of the particular solution.
C27 Example problem finding the form of the particular solution.
C28 Variation of parameters Part 1.
C29 Variation of parameters Part 2.
C30 Solving a linear DE by the annihilator approach.
C31 The same problem but using variation of parameters.
C32 Example problem using variation of parameters.
C33 Example problem using variation of parameters.
C34 Expanding this method to higher order linear differential equations.
C35 The Cauchy Euler Equation.
C36 Example problem solving a Cauchy Euler equation.
C37 Example problem solving a Cauchy Euler equation.
C38 Example problem solving a Cauchy Euler equation with initial values.
C39 A Cauchy Euler equation that is nonhomogeneous.
C40 Example problem solving a nonhomogeneous Cauchy Euler equation.
C41 Using substitution to solve a Cauchy Euler equation.
C42 Example problem solving a Cauchy Euler equation with substitution.
C43 Example problem solving a Cauchy Euler equation.
C44 Example problem solving a Cauchy Euler equation.
C45 Example problem solving a Cauchy Euler equation.
C46 Solving the previous problem by another method.
C47 Example problem solving a Cauchy Euler equation.
C48 Systems of linear differential equations.
C49 Example problem solving a system of linear DEs Part 1.
C50 Example problem solving a system of linear DEs Part 2.
C51 Example problem of a system of linear DEs.
C52 Introduction to nonlinear DEs.
C53 Introduction to modelling.
C54 Free undamped motion.
C55 Example problem of free undamped motion.
C56 Continuation of previous problem.
C57 Alternate form of the solution.
C58 Example problem using the alternate form.
C59 Free damped motion.
C60 Example problem involving free damped motion.
C61 Another problem involving free undamped motion.
C62 Driven motion with damping.
C63 Example problem involving driven motion with damping.
C64 Transient and steady state terms.
C65 Exampe problem illustrating transient and steady state terms.
C66 Resonance.
C67 The physics of simple harmonic motion.
C68 The physics of damped motion.
C69 Introduction to power series.
C70 The ratio test for power series.
C71 Adding to power series in sigma notation.
C72 What to do about the singular point.
C73 Introducing the theorem of Frobenius.
C74 Example problem.
C75 Introduction to the Laplace Transform.
C76 A first example problem calculating the Laplace transform.
C77 Another example problem calculating the Laplace transform.
C78 Another example problem calculating the Laplace transform.
C79 Linear properties of the Laplace transform.
C80 Solving a linear DE with Laplace transformations.
C81 More complex Laplace tranformations.
C82 Example problem using inverse Laplace transforms Part 1.
C83 Example problem using inverse Laplace transforms Part 2.
C84 Example problem using inverse Laplace transforms Part 3.
Taught by
Dr Juan Klopper
