Differential Equations Cheat Sheet

Updated 2026-04-28

Next Topic: Discrete Mathematics Cheat Sheet

Differential equations are mathematical equations that relate a function to its derivatives, forming the backbone of mathematical modeling in physics, engineering, biology, and economics. They describe how quantities change continuously in response to various factors—whether modeling population growth, heat distribution, fluid dynamics, or electrical circuits. The key distinction lies between ordinary differential equations (ODEs), involving derivatives with respect to a single variable, and partial differential equations (PDEs), involving multiple variables. Understanding solution techniques—from separation of variables to numerical methods—enables practitioners to predict system behavior, analyze stability, and solve real-world problems that would be intractable through algebraic methods alone.

What This Cheat Sheet Covers

This topic spans 17 focused tables and 152 indexed concepts. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.

Table 1: Classification and TypesTable 2: First-Order ODE Solution MethodsTable 3: Second-Order Linear ODE MethodsTable 4: Fundamental Concepts and TheoryTable 5: Laplace Transform MethodsTable 6: Series SolutionsTable 7: Systems of Differential EquationsTable 8: Stability AnalysisTable 9: Bifurcation TheoryTable 10: Numerical Methods for ODEsTable 11: Partial Differential Equations (PDEs)Table 12: Boundary Conditions for PDEsTable 13: Advanced Solution TechniquesTable 14: Special Functions and Classical EquationsTable 15: Applications and Physical InterpretationsTable 16: Numerical Methods for PDEsTable 17: Stochastic, Delay, and Fractional Equations

Table 1: Classification and Types

Type	Example	Description
Ordinary Differential Equation (ODE)	$\frac{dy}{dx} + 3y = e^x$	• Contains derivatives with respect to only one independent variable • contrasts with PDEs which have multiple variables.
Partial Differential Equation (PDE)	$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$	• Involves partial derivatives with respect to two or more independent variables • models phenomena in multiple dimensions.
First-Order Equation	$y' = 2x + y$	• Contains only the first derivative as the highest-order derivative • requires one initial condition for unique solution.
Second-Order Equation	$y'' - 5y' + 6y = 0$	• Involves the second derivative as the highest-order derivative • requires two initial conditions for unique solution.
Higher-Order Equation	$y''' + y'' - y' + y = 0$	• Contains derivatives of order three or higher • order equals the highest derivative present.
Linear Differential Equation	$y'' + p(x)y' + q(x)y = g(x)$	• The dependent variable and its derivatives appear to the first power only • no products or nonlinear functions of $y$ or its derivatives.

Table 1: Classification and Types

Type	Example	Description
Ordinary Differential Equation (ODE)	$\frac{dy}{dx} + 3y = e^x$	• Contains derivatives with respect to only one independent variable • contrasts with PDEs which have multiple variables.
Partial Differential Equation (PDE)	$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$	• Involves partial derivatives with respect to two or more independent variables • models phenomena in multiple dimensions.
First-Order Equation	$y' = 2x + y$	• Contains only the first derivative as the highest-order derivative • requires one initial condition for unique solution.
Second-Order Equation	$y'' - 5y' + 6y = 0$	• Involves the second derivative as the highest-order derivative • requires two initial conditions for unique solution.
Higher-Order Equation	$y''' + y'' - y' + y = 0$	• Contains derivatives of order three or higher • order equals the highest derivative present.
Linear Differential Equation	$y'' + p(x)y' + q(x)y = g(x)$	• The dependent variable and its derivatives appear to the first power only • no products or nonlinear functions of $y$ or its derivatives.