Ordinary Differential Equations Course: Free Videos and Guided Notes

Master ordinary differential equations through structured video lessons and guided notes. This free online differential equations course covers first-order differential equations, second-order differential equations, Laplace transforms, systems of differential equations, and mathematical modeling while building understanding and problem-solving skills step-by-step.

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7 Completed Notes

Chapter 1 - Introduction to Differential Equations

Watch the video lessons for each section.
Download the guided notes PDF to follow along as you watch.
Engage actively by completing the notes during the lesson.
Review your completed notes as a study resource for exams.

In this video, we classify differential equations by type, order, and linearity—a foundational skill in any differential equations course.
▶️ Video‍ ‍📝 Notes‍

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Chapter 2 - First-Order Differential Equations

Watch the video lessons for each section.
Download the guided notes PDF to follow along as you watch.
Engage actively by completing the notes during the lesson.
Review your completed notes as a study resource for exams.

In this video, we introduce slope fields (also called direction fields) for first-order differential equations and show how they are used to visualize solution curves.
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In this video, we introduce autonomous first-order differential equations and show how to analyze their behavior using critical points, phase portraits, and slope fields.
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In this video, we introduce separable differential equations and show how to solve them using the method of separation of variables—one of the most important techniques in a first course on differential equations
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In this video, we introduce first-order linear differential equations and show how to solve them using the integrating factor method—a core technique in any differential equations course.
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In this video, we introduce exact differential equations and show how to solve them using a systematic, step-by-step process.
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In this video, we introduce homogeneous differential equations and show how to solve them using a systematic change of variables.
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In this video, we introduce Bernoulli differential equations and show how to solve them by reducing the equation to a first-order linear differential equation using a substitution.
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In this video, we apply first-order linear differential equations to several classic real-world modeling problems.
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In this video, we introduce Euler’s Method, a numerical technique used to approximate solutions to first-order ordinary differential equations when an exact solution is difficult or impossible to find.
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In this video, we study the logistic differential equation, a first-order separable differential equation used to model population growth with a carrying capacity.
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In this video, we work through an exam review covering classification and first-order ordinary differential equations. The video is perfect for studying for midterms or a final exam.
▶️ Video ‍📝 Notes‍

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Chapter 3 - Second-Order Differential Equations

Watch the video lessons for each section.
Download the guided notes PDF to follow along as you watch.
Engage actively by completing the notes during the lesson.
Review your completed notes as a study resource for exams.

In this video, we study homogeneous linear differential equations with constant coefficients, focusing primarily on second-order differential equations and extending the method to higher-order equations.
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In this video, we study Reduction of Order, a method used to find a second, linearly independent solution to a second-order linear differential equation when one solution is already known.
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In this video, we study the Method of Undetermined Coefficients, a standard technique used to solve second-order constant coefficient nonhomogeneous differential equations of the form: ay″ + by′ + cy = g(x) where g(x) is a polynomial, exponential, sine or cosine function, or a combination of these functions.
▶️ Video‍ ‍📝 Notes‍

In this video, we study the Method of Variation of Parameters, a powerful technique used to solve second-order nonhomogeneous differential equations of the form: ay″ + by′ + cy = g(x) for any function g(x).
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In this video, we study second-order homogeneous Cauchy–Euler differential equations, a special class of variable coefficient differential equations of the form: a x²y'' + b x y' + c y = 0.
▶️ Video‍ ‍📝 Notes‍

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Chapter 4 - Differential Equations Applications and Modeling

Watch the video lessons for each section.
Download the guided notes PDF to follow along as you watch.
Engage actively by completing the notes during the lesson.
Review your completed notes as a study resource for exams.

In this video, we derive and solve the differential equation that models a mass–spring system with no damping and no external forcing.
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In this video, we model and solve a mass–spring system with damping and no external forcing.
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In this video, we model and solve a mass–spring system with damping and external forcing.
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In this video, we solve a boundary value problem and introduce the concepts of eigenvalues and eigenfunctions in differential equations.
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Chapter 5 - Series Solutions

Watch the video lessons for each section.
Download the guided notes PDF to follow along as you watch.
Engage actively by completing the notes during the lesson.
Review your completed notes as a study resource for exams.

In this video, we solve a differential equation using a power series centered at x=0 and walk through the entire method step by step.
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In this video, we use the power series method to solve a differential equation and work through Airy’s equation step by step
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In this video, we learn how to identify and classify ordinary and singular points of a differential equation.
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In this video, we use the method of Frobenius to solve a second-order differential equation with a regular singular point (Bessel's equation of order zero).
▶️ Video ‍📝 Notes‍

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Chapter 6 - Laplace Transforms

Watch the video lessons for each section.
Download the guided notes PDF to follow along as you watch.
Engage actively by completing the notes during the lesson.
Review your completed notes as a study resource for exams.

In this video, we introduce the Laplace transform and explain how it is used to simplify differential equations.
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In this video, we introduce the Laplace transform and explain how it is used to simplify differential equations.
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In this video, we learn how to take the Laplace transform of derivatives and see how differential equations become algebraic equations.
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In this video, we learn how to find inverse Laplace transforms using a table, properties, and step-by-step techniques.
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In this video, we solve an initial value problem using Laplace transforms step by step.
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In this video, we introduce step functions, translation (shifting), and delta functions, and show how they are used with Laplace transforms to solve differential equations.
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In this video, we introduce convolution and show how it is used with Laplace transforms to evaluate inverse transforms and solve differential equations.
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Chapter 7 - Systems of Differential Equations

Watch the video lessons for each section.
Download the guided notes PDF to follow along as you watch.
Engage actively by completing the notes during the lesson.
Review your completed notes as a study resource for exams.

In this video, we show how to rewrite a second-order differential equation as a system of first-order equations.
▶️ Video ‍📝 Notes‍
In this video, we solve systems of differential equations using eigenvalues. We walk through step-by-step examples covering distinct real eigenvalues, repeated eigenvalues, and complex eigenvalues.
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In this video, we find and classify critical points of autonomous systems of differential equations using eigenvalues.
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In this video, we learn how to linearize nonlinear systems of differential equations near critical points using the Jacobian matrix.
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About Dr. Cheryl Hile

Understand The Math was created by Dr. Cheryl Hile, who earned a Ph.D. in Engineering Science and Applied Mathematics from Northwestern University. She has more than 25 years of experience teaching mathematics at the university level, including 20 years at Penn State. The lessons and guided notes on this site are based on materials she developed for her university courses.