Differential Equations A Problem Based Approach

Name: Differential Equations A Problem Based Approach
Rating: 4.9 (2 reviews)

Solving Ordinary Differential Equations like there is no tomorrow

Created byDr. Ron Erez

Last updated 12/2025

English

What you'll learn

Solve Systems of First-Order Linear ODEs
Python for visualizing the topics we will learn
Linear algebra background needed for solving differential equations
Calculus background needed for solving differential equations
Apply the Theory of Constant Coefficients
Solve Systems of First-Order Linear ODEs

Course content

9 sections • 58 lectures • 3h 52m total length

Introduction0:53
Explore ordinary differential equations through a problem-based approach, using practical problem solving techniques and essential calculus and linear algebra concepts, with weekly problems and detailed solutions.

Understanding Ordinary Differential Equations- What They Are and Why They Matter6:26
Linear First Order Differential Equations10:36
Finding the Solution to cos(x) * y' + sin(x) * y = 05:04
Solving y' + y = xe^x - Integrating Factor5:48
Use the integrating factor e^x to solve y' + y = x e^x, convert to derivative of y e^x, and apply integration by parts.
Solving y' + y = xe^x - Integration by Parts4:53
Solving y' + y = xe^x - Completing the Solution3:35

Example of a Separable ODE2:48
Exercise 1 - Solving a Separable ODE of the Form a(x,y)dx+b(x,y)dy=03:25
Solve a separable first-order ode by separating variables, integrating to relate ln(x^2+1) with 1/y, and obtain the solution y = 1 / (1/2 ln(x^2+1) + c).
Exercise 2 - Initial Value Problem0:34
Solution 2 - Initial Value Problem9:45
Exercise and Solution 3 - Change Initial Condition and Asymptotics9:37

Bernoulli ODEs - Theory7:43
Exercise 1 - Bernoulli ODE6:35
Exercise 1 - Bernoulli ODE - Reduce to Separation of Variables2:44
Exercise 1 - Bernoulli ODE - Finding y using v4:12
Exercise 2 - Bernoulli ODE and Initial Conditions1:02
Solve an initial value problem by rewriting the ode as a Bernoulli equation and using an integrating factor. Determine the constant from the initial condition and finish the solution.
Exercise 2 - Bernoulli ODE - Integrating Factor3:24
Solve a Bernoulli ode by applying an integrating factor e^{-x}, transform with v = y e^{-x}, and use separation of variables to complete the exercise.
Exercise 2 - Bernoulli ODE - Reducing to Separation of Variables2:56
Solution 2 - Bernoulli ODE - Integration by Parts4:31
Solution 2 - Bernoulli ODE - Completing Solution1:31

Constant Coefficient ODEs - Introducinng the Characteristic Equation3:49
Constant Coefficient ODEs - Contribution of the Algebraic Multiplicity5:23
Explore how algebraic multiplicity of roots in the characteristic polynomial yields additional solutions for constant-coefficient linear ODEs, by multiplying e^{λx} by powers of x to complete the fundamental solution set.
Characteristic Equations - Handling Complex Roots3:13
Characteristic Equations - Obtaining Real Solutions from Complex Roots3:14
Higher-Order ODEs with Characteristic Polynomials and Polynomial Division6:12
Solve third-order homogeneous linear differential equation with constant coefficients by factoring its characteristic polynomial, yielding roots 2, 7, -4, and y = c1 e^{2x} + c2 e^{7x} + c3 e^{-4x}.
Linear Inhomogeneous Second-Order ODEs Using Linear Combinations8:31
Solving a Separable ODE of the Form a(x,y)dx + b(x,y)dy=03:25
Solve a separable first-order ode by separating variables and integrating. Obtain y = 1 / ( (1/2) ln(x^2+1) + C ), using the 1/y^2 integral.
Solving Higher-Order ODEs with Complex Roots of the Characteristic Equation5:07

Introduction to ay'' + by' + cy = 01:27
The Characteristic Equation1:30
Learn to form the characteristic equation from a linear differential equation with e^{r x}, divide by e^{r x}, and classify roots via b^2 - 4ac into real, repeated, or complex.
THEORY - Solving ay'' + by' + cy = 02:10
WORKSHEET - Exercises of the form ay'' + by' + cy = 01:19
Solve the worksheet on second order linear homogeneous ODEs ay'' + by' + cy = 0, finding the characteristic equation, roots, the general solution, and initial conditions when given.
SOLUTION - PROBLEM 11:54
solve the differential equation y'' + y' = 0 via the characteristic equation r(r+1)=0, yielding y(x) = c1 + c2 e^x, as part of problem-based learning.
SOLUTION - PROBLEM 23:22
SOLUTION PROBLEM 32:40
SOLUTION PROBLEM 41:43

What is a System of ODEs?1:56
Matrix Notation Example for Linear ODE Systems with Constant Coefficient2:43
Converting a Linear ODE to a System of First Order ODEs2:47
Convert a linear nth-order ODE to a first-order system by defining y0 = y, y1 = y', ..., y(n-1) = y^(n-1), and express y' = A y.
Structure of Solution Sets for First-Order Linear ODE Systems3:21
Systems of First-Order ODEs with Constant Coefficients2:51
Explore solving systems of first-order odes with constant coefficients by seeking x = v e^{λ t} and deriving the eigenvalue–eigenvector condition A v = λ v.
EXERCISE First-Order ODE Systems with Constant Coefficients Using Linear Algebra7:20
EXERCISE Finding the Eigenvectors for Lambda = 25:36
EXERCISE Finding the Eigenvectors for Lambda = -14:10
EXERCISE Solving First-Order ODE Systems with Complex Eigenvalues4:24
Theory - Inhomogeneous ODE Systems and Variation of Parameters4:57
Solving Inhomogeneous - Deriving a Formula for c'(t)3:48
Solving Inhomogeneous - Integrating c'(t)1:48

Requirements

Basic knowledge of algebra and calculus (e.g., derivatives, integrals, exponential functions, basic trig)
Familiarity with functions and graphs (You'll be visualizing solutions to differential equations)
Introductory Linear Algebra (Matrix multiplication, determinants, solving systems of equations, and working with vectors)
Note: We’ll recall and refresh all the necessary mathematical background as we go along. And if anything is unclear, you can always reach out to the instructor for further clarification.

Description

Unlock the power of differential equations through clear, example-driven instruction designed for beginners and beyond. This comprehensive course walks you step-by-step through solving first-order and higher-order ordinary differential equations (ODEs), with a strong focus on intuition, methodical techniques, and real-world applications.

You'll start with the fundamentals: first-order equations and the Existence and Uniqueness Theorem. From there, we move into second-order linear equations, exploring homogeneous equations, linear independence, the Wronskian, and techniques like lowering the order of an equation. You'll master solving equations with constant coefficients and learn how to separate problems into homogeneous and inhomogeneous parts.

Key solution methods—including the Method of Undetermined Coefficients and Variation of Parameters—are explained through worked examples. The course generalizes to nth-order equations and includes Euler's formula and a deep dive into the powerful Laplace Transform, including initial/final value theorems and convolution applications.

You'll also explore systems of first-order linear equations using matrix methods and linear algebra, and gain insight into Sturm-Liouville problems and self-adjoint operators, with a focus on eigenfunctions and eigenvalues—crucial tools in physics and engineering.

Whether you're a student, engineer, data scientist, or self-learner, this course will equip you with the tools and confidence to solve differential equations in both academic and applied settings.

Who this course is for:

Anybody who would like to learn how to solve differential equations
Students of the Sciences — including Engineering, Physics, Biology, Mathematics, and Economics
College and University Students taking STEM or quantitative courses
Professionals who need to use differential equations for modeling or analysis in engineering, finance, biology, or research
Self-learners who want a straightforward, no-nonsense approach to learning math
This course is accessible, yet rigorous. It’s for learners who want to get to the point, build intuition, and actually solve problems — not just watch theory fly by.

Differential Equations A Problem Based Approach

What you'll learn

Explore related topics

Course content

Introduction1 lecture • 1min

What is an ODE and First Order Differential Equations6 lectures • 36min

Solving Nonlinear Separable ODEs5 lectures • 26min

Homogeneous first-order ODEs5 lectures • 23min

Nonlinear Bernoulli ODEs9 lectures • 35min

Linear Ordinary Differential Equations8 lectures • 39min

Second order ODEs8 lectures • 16min

Second Order Inhomogeneous ODEs using the Method of Variation of Parameters4 lectures • 11min

Systems of ODEs12 lectures • 46min

Requirements

Description

Who this course is for: