“All the world’s a differential equation,
and the men and women are merely variables.”

Ben Orlin, Change is the Only Constant: The Wisdom of Calculus in a Madcap World

## Definition

differential equation is an equation which contains one or more terms and the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable)

dy/dx = f(x)

Here “x” is an independent variable and “y” is a dependent variable

For example, dy/dx = 5x

A differential equation contains derivatives which are either partial derivatives or ordinary derivatives. The derivative represents a rate of change, and the differential equation describes a relationship between the quantity that is continuously varying with respect to the change in another quantity. There are a lot of differential equations formulas to find the solution of the derivatives.1

## Why

In Mathematics, a differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on. The primary purpose of the differential equation is the study of solutions that satisfy the equations and the properties of the solutions. Learn how to solve differential equations here.1

## How

Many of the References and Additional Reading websites and Videos will assist you with understanding differential equations.

As some professors say: “It is intuitively obvious to even the most casual observer.

1 “Differential Equations (Definition, Types, Order, Degree, Examples)”. 2022. BYJUS. https://byjus.com/maths/differential-equation/.

Bronson, Richard. “Schaum’s Easy Outline: Differential Equations”. 2003. math24.files.wordpress.com. https://math24.files.wordpress.com/2016/04/shuam-ode-book.pdf.

“Differential Equations”. 2022. tutorial.math.lamar.edu. https://tutorial.math.lamar.edu/classes/DE/DE.aspx.

“DIFFERENTIAL EQUATIONS”. 2022. ncert.nic.in. https://ncert.nic.in/textbook/pdf/lemh203.pdf.

“Differential Equations – Introduction”. 2022. mathsisfun.com. https://www.mathsisfun.com/calculus/differential-equations.html.

“Differential Equations Solution Guide”. 2022. mathsisfun.com. https://www.mathsisfun.com/calculus/differential-equations-solution-guide.html.

“Introduction to Differential Equations”. 2022. personal.psu.edu. http://www.personal.psu.edu/bwo1/courses/Dennis/section1-1.pdf.

Logan, J. David. “A First Course in Differential Equations”. 2015. math.unl.edu. https://www.math.unl.edu/~jlogan1/PDFfiles/New3rdEditionODE.pdf.

## Videos

Differential equations are usually classified into two general categories: partial differential equations, which are also called partial derivatives, and ordinary differential equations. Partial derivatives can be first-order, second-order, or higher order, and they’re what you get when you have a multivariable equation, but take the derivative of that equation with respect to just one variable. Ordinary differential equations are equations that include a derivative, and that derivative can be of any order (first-order, second-order, or higher-order). Ordinary differential equations are more commonly classified into other specific types of differential equations, like linear differential equations, separable differential equations, and exact differential equations.

“Vector Transformation Visualization Tool (Vtvt) — Another Demo”. 2019. Lineae Ex Punctis. https://www.expunctis.com/2019/04/04/vtvt-another-demo.html.

This demo visualizes numerical solutions of the pendulum equation by 3blue1brown (14:28) in the above video.

## Articles by Spencer Hooks

“Differential Equations 0: Complex Numbers, Roots of Polynomial Equations, & Elements of Matrix Theory”. 2021. Medium. https://medium.com/intuition/differential-equations-0-complex-numbers-roots-of-polynomial-equations-elements-of-matrix-b2b21e4c7522.

This article covers the prerequisites and basics to be used throughout this series of articles. It is a long read but you can just skip to the theorems and take notes of them.

“Differential Equations 1: First-Order Differential Equations”. 2021. Medium. https://medium.com/intuition/differential-equations-1-first-order-differential-equations-b9ca2dfc1a30.

Differential equations play a vital role in the solution of many problems encountered when modelling physical phenomena. All the disciplines in the physical sciences, each with its unique physical situations, require that you can derive the necessary differential equations and then solve them. This article covers the definitions and basic characteristics of differential equations.

“Differential Equations 2: First-Order Linear Equations”. 2021. Medium. https://medium.com/intuition/differential-equations-2-first-order-linear-equations-2153cbf02647.

This article covers the different types of differential equations, theorems, and corollaries.

“Differential Equations 3: Applications Of First-Order Linear Equations”. 2021. Medium. https://medium.com/intuition/differential-equations-3-applications-of-first-order-linear-equations-1db29472b90a.

This article covers a couple of the many applications of first-order linear equations.

“Differential Equations 4: Nonlinear Equations Of First Order”. 2021. Medium. https://medium.com/intuition/differential-equations-4-nonlinear-equations-of-first-order-94b8c8542490.

This article covers the nonlinear equations of first order, separable equations, and exact equations.

“Differential Equations 5: Linear Systems With Constant Coefficients”. 2021. Medium. https://medium.com/intuition/differential-equations-5-linear-systems-with-constant-coefficients-f1fcc55e7d5b.

This article covers an introduction to some notation, eigenvectors and eigenvalues, and some systems of linear, first-order, homogeneous differential equations.

“Natura non facit saltum or, Nature does not make leaps… If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future).”

Benoit Mandelbrot

“Differential Equations 6: Complex Eigenvalues, Repeated Eigenvalues, & Fundamental Solution Matrices”. 2021. Medium. https://medium.com/intuition/differential-equations-6-complex-eigenvalues-repeated-eigenvalues-fundamental-solution-c9048936c2a1.

This article covers complex eigenvalues, repeated eigenvalues, & fundamental solution matrices, plus a small look into using the Laplace transform in the future to deal with fundamental solution matrices.

“Differential Equations 7: Fundamental Theorems, Solutions Of Nonhomogeneous Systems, & Undetermined Coefficients”. 2021. Medium. https://medium.com/intuition/differential-equations-7-fundamental-theorems-solutions-of-nonhomogeneous-systems-undetermined-656b1230775b.

This article covers some fundamental theorems, solutions of nonhomogeneous systems, and undetermined coefficients.

“Differential Equations 8: Variation of Parameters, Nonhomogeneous Initial-Value Problems, & More on Fundamental Solution Matrices”. 2021. Medium. https://medium.com/intuition/differential-equations-8-variation-of-parameters-nonhomogeneous-initial-value-problems-more-on-f22a048c572.

This article covers the variation of parameters method, nonhomogeneous initial-value problems, and more on fundamental solution matrices.

“Differential Equations 9: Second-Order Linear Equations”. 2021. Medium. https://medium.com/intuition/differential-equations-9-second-order-linear-equations-49a59c866f32.

This article covers a short introduction and sectionally continuous functions, linear differential operators, and linear independence and the Wronskian.

“Differential Equations 10: The Nonhomogeneous Equation & Constant Coefficient Homogeneous Equations”. 2021. Medium. https://medium.com/intuition/differential-equations-10-the-nonhomogeneous-equation-constant-coefficient-homogeneous-equations-7e5e18e828d6.

This article covers the nonhomogeneous equation, constant-coefficient homogeneous equations and the real and unequal roots, the real and equal roots, and the complex roots.

“Mathematics is not about numbers, equations, computation,
or algorithms: it is about understanding.”

William Thurston

“Differential Equations 11: Spring-Mass Systems In Free Motion, Undamped Motion, & Damped Motion”. 2021. Medium. https://medium.com/intuition/differential-equations-11-spring-mass-systems-in-free-motion-undamped-motion-damped-motion-a5dad3630231.

This article covers spring-mass systems in free motion, undamped motion, & damped motion.

“Differential Equations 12: The Electric Circuit & Undetermined Coefficients”. 2021. Medium. https://medium.com/intuition/differential-equations-12-the-electric-circuit-undetermined-coefficients-d77d3eb7aa69.

Consider a circuit containing a resistance R, inductance L, and capacitance C in series. This circuit is the same one as a previous article. In the previous article by omitting selected elements we showed that the equation governing the voltage and current reduced to a first-order ODE. In this article, we will study the same circuit in a more general context.

“Differential Equations 13: The Spring-Mass System: Forced Motion”. 2022. Medium. https://medium.com/intuition/differential-equations-13-the-spring-mass-system-forced-motion-c78fa83b843e.

“Differential Equations 14: The Cauchy-Euler Equation & Variation Of Parameters”. 2022. Medium. https://medium.com/intuition/differential-equations-14-the-cauchy-euler-equation-variation-of-parameters-49faed2f4aec.

“Differential Equations 15: Higher Order Equations”. 2022. Medium. https://medium.com/intuition/differential-equations-15-higher-order-equations-7e156a446eb1.

“Mathematics is the cheapest science. Unlike physics or chemistry
it does not require any expensive equipment.
All one needs for mathematics is a pen and paper.”

George Polya

“Differential Equations 16: The Nonhomogeneous Equation & Companion Systems”. 2022. Medium. https://medium.com/intuition/differential-equations-16-the-nonhomogeneous-equation-companion-systems-f4c7e345a971.

“Differential Equations 17: Companion Systems & Initial-Value Problems for Homogeneous Companion Systems”. 2022. Medium. https://medium.com/intuition/differential-equations-17-companion-systems-initial-value-problems-for-homogeneous-companion-479aebb737b3.

“Differential Equations 18: Variation Of Parameters For Nonhomogeneous Companion Systems”. 2022. Medium. https://medium.com/intuition/differential-equations-18-variation-of-parameters-for-nonhomogeneous-companion-systems-935ff0fe0620.

“Differential Equations 19: The Laplace Transform & Its General Properties”. 2022. Medium. https://medium.com/intuition/differential-equations-19-the-laplace-transform-its-general-properties-b30a899afb82.

“Differential Equations : Sectionally Continuous Functions & Laplace Transforms Of Periodic…”. 2022. Medium. https://medium.com/intuition/differential-equations-sectionally-continuous-functions-laplace-transforms-of-periodic-2471fa047aee.

Newton has shown us that a law is only a necessary relation between the present state of the world and its immediately subsequent state. All the other laws since discovered are nothing else; they are in sum, differential equations.

Henri Poincaré

“Differential Equations 21: The Inverse Laplace Transform, Partial Fractions, & Nonrepeated Linear Factors”. 2022. Medium. https://medium.com/intuition/differential-equations-21-the-inverse-laplace-transform-partial-fractions-nonrepeated-linear-b3438108e2fd.

“Differential Equations 23: The Solution Of Initial-Value Problems & The Laplace Transform Of Systems”. 2022. Medium. https://medium.com/intuition/differential-equations-23-the-solution-of-initial-value-problems-the-laplace-transform-of-4e653ee79be9.

“Differential Equations 24: Series Methods & Analytic Functions”. 2022. Medium. https://medium.com/intuition/differential-equations-24-series-methods-analytic-functions-a8a8f4e1547e.

“Differential Equations 25: Taylor Series Of Analytic Functions”. 2022. Medium. https://medium.com/intuition/differential-equations-25-taylor-series-of-analytic-functions-8a9874507360.

It is a curious historical fact that modern quantum mechanics began with two quite different mathematical formulations: the differential equation of Schroedinger and the matrix algebra of Heisenberg. The two apparently dissimilar approaches were proved to be mathematically equivalent.

Richard P. Feynman

“Differential Equations 26: Power Series Solutions Of Linear Differential Equations”. 2022. Medium. https://medium.com/intuition/differential-equations-26-power-series-solutions-of-linear-differential-equations-32281ca75418.

“Differential Equations 27: Power Series Solutions Of Linear Differential Equations Pt. 2”. 2022. Medium. https://medium.com/intuition/differential-equations-27-power-series-solutions-of-linear-differential-equations-pt-2-2dd898b61dd0.

“Differential Equations 28: Legendre’s Equation”. 2022. Medium. https://medium.com/intuition/differential-equations-28-legendres-equation-741eb2da04bd.

“Differential Equations 29: Three Important Application Examples”. 2022. Medium. https://medium.com/intuition/differential-equations-29-three-important-application-examples-db5143aaac92.

“Differential Equations 30: Bessel’s Equation”. 2022. Medium. https://medium.com/intuition/differential-equations-30-bessels-equation-452e55548ce0.

“Differential Equations 31: The Wronskian Method”. 2022. Medium. https://medium.com/intuition/differential-equations-31-the-wronskian-method-de19e961af3.

“Differential Equations 32: The Frobenius Method”. 2022. Medium. https://medium.com/intuition/differential-equations-32-the-frobenius-method-81d758ca85ce.

“Differential Equations 33: Direction Fields”. 2022. Medium. https://medium.com/intuition/differential-equations-33-direction-fields-3ee2e0685e67.