About Differential Equations

Differential equations are the natural extension of everything you’ve learned in calculus. While derivatives tell you how fast something is changing at a single moment, differential equations let you predict how that thing will behave over time. In this subject you’ll learn how to solve them, visualize their behavior, and apply them to real phenomena.

A differential equation is an equation that relates a function to one or more of its derivatives. In other words, it describes how a quantity changes in relation to itself or other quantities.

The simplest example is:

$$\frac{dy}{dx} = ky$$

This says “the rate of change of $y$ is proportional to $y$ itself.” Solutions to this equation are exponential functions, the foundation of population growth, radioactive decay, and compound interest.

Differential equations come in many flavors: first-order, second-order, linear, nonlinear, ordinary, and partial. We’ll focus primarily on ordinary differential equations (ODEs) in this course.

The visual above shows solution curves flowing through a direction field, one of the most powerful ways to understand differential equations before you even solve them.

Differential equations emerged alongside calculus itself in the late 1600s. Isaac Newton used them to describe planetary motion, while Leibniz developed much of the notation we still use today.

In the 1700s and 1800s, mathematicians like Euler, Bernoulli, Lagrange, and Laplace turned differential equations into a systematic field. The Laplace transform (which you’ll learn later) was revolutionary for solving complex problems in engineering and physics. By the 20th century, differential equations became essential for modeling everything from electrical circuits to quantum mechanics and climate systems.

You’ve already built strong foundations in Calculus 1-3 and Linear Algebra. Differential equations are where those tools truly come alive. Derivatives and integrals stop being abstract operations and start describing real behavior over time. Linear algebra gives you the tools to handle systems of equations. This subject ties everything together and opens the door to advanced work in physics, engineering, computer graphics, biology, economics, and machine learning.

• Engineering: Modeling vibrations in bridges, circuit behavior, and control systems
• Physics: Describing motion of springs, pendulums, orbiting bodies, and heat flow
• Biology and Medicine: Population dynamics, spread of diseases, drug concentration in the bloodstream
• Economics and Finance: Growth models, interest rates, and market behavior
• Computer Graphics and Games: Realistic physics for cloth, water, particles, and character movement
• Machine Learning: Training neural networks (gradient descent is essentially solving a differential equation)

Foundations Direction fields, separable equations, first-order linear, and exact equations.

Classic Applications Exponential growth/decay, Newton’s Law of Cooling, mixing problems.

Second-Order Equations Homogeneous and nonhomogeneous equations, undetermined coefficients, variation of parameters, and harmonic motion.

Advanced Tools Laplace transforms for solving initial value problems with ease.

Systems and Numerical Methods Systems of differential equations, phase plane analysis, and Euler’s method when exact solutions aren’t possible.

You should be comfortable with:

• Derivatives and basic integration (Calculus 1)
• Integration techniques including integration by parts and partial fractions (Calculus 2)
• Vectors, matrices, and eigenvalues (Linear Algebra)

If you’ve completed up through Calculus 3 and Linear Algebra here at Mom’s Basement University, you’re well prepared.

Sketch direction fields by hand. It builds intuition faster than any formula. Work through every example step-by-step before looking at the solution. When you get stuck on a harder problem, go back to the direction field or try reducing it to a simpler case. Play with the interactive visuals. And most importantly: connect every technique back to a real-world situation. That’s where the real understanding happens.