Differential Equations Basics – Complete Cheat Sheet

Introduction

Differential equations are mathematical equations that relate functions to their derivatives. They describe how quantities change over time or space and are fundamental to modeling real-world phenomena in physics, engineering, biology, economics, and other fields. Understanding differential equations is crucial for analyzing dynamic systems and predicting future behavior based on current conditions and rates of change.

Core Concepts & Principles

What is a Differential Equation?

A differential equation contains one or more derivatives of an unknown function. The goal is to find the function that satisfies the equation.

General Form: F(x, y, y’, y”, …, y⁽ⁿ⁾) = 0

Key Terminology

Classification System

By Order:

• First Order: Contains only dy/dx
• Second Order: Contains d²y/dx²
• Higher Order: Contains derivatives of order 3 and above

By Linearity:

• Linear: a₁(x)y⁽ⁿ⁾ + a₂(x)y⁽ⁿ⁻¹⁾ + … + aₙ(x)y = f(x)
• Nonlinear: Contains products, powers, or transcendental functions of y

Step-by-Step Solution Process

General Problem-Solving Approach

1. Identify the Type

   • Determine order, degree, and linearity
   • Check if separable, exact, or linear

2. Choose the Method

   • Match equation type to appropriate solution technique
   • Consider initial/boundary conditions

3. Apply the Technique

   • Follow method-specific steps
   • Perform necessary integrations

4. Find General Solution

   • Include all arbitrary constants
   • Verify by substitution

5. Apply Initial Conditions

   • Use given conditions to find specific constants
   • Write particular solution

6. Verify Solution

   • Substitute back into original equation
   • Check initial conditions are satisfied

Key Solution Techniques by Category

First-Order Differential Equations

Separable Equations

Form: dy/dx = f(x)g(y)

Method:

1. Separate variables: dy/g(y) = f(x)dx
2. Integrate both sides: ∫dy/g(y) = ∫f(x)dx
3. Solve for y

Example: dy/dx = xy

• Solution: dy/y = x dx → ln|y| = x²/2 + C → y = Ae^(x²/2)

Linear First-Order Equations

Form: dy/dx + P(x)y = Q(x)

Method (Integrating Factor):

1. Find integrating factor: μ(x) = e^(∫P(x)dx)
2. Multiply equation by μ(x)
3. Left side becomes d/dx[μ(x)y]
4. Integrate: μ(x)y = ∫μ(x)Q(x)dx

Exact Equations

Form: M(x,y)dx + N(x,y)dy = 0

Condition: ∂M/∂y = ∂N/∂x

Method:

1. Verify exactness condition
2. Find function F(x,y) where ∂F/∂x = M and ∂F/∂y = N
3. Solution: F(x,y) = C

Second-Order Differential Equations

Homogeneous Linear with Constant Coefficients

Form: ay” + by’ + cy = 0

Method (Characteristic Equation):

1. Assume solution y = e^(rx)
2. Substitute to get: ar² + br + c = 0
3. Solve for roots r₁, r₂

Solution Types:

Non-homogeneous Linear Equations

Form: ay” + by’ + cy = f(x)

Method: y = y_h + y_p (homogeneous + particular solution)

Particular Solution Methods:

Comparison of Solution Methods

Common Challenges & Solutions

Challenge 1: Identifying Equation Type

Problem: Difficulty classifying equations Solution:

• Create a checklist: order, linearity, coefficients
• Practice with varied examples
• Look for standard forms

Challenge 2: Integration Difficulties

Problem: Complex integrals during solution Solution:

• Review integration techniques
• Use substitution methods
• Consider numerical approaches for complex cases

Challenge 3: Applying Initial Conditions

Problem: Incorrectly using given conditions Solution:

• Carefully identify what’s given (y(x₀), y'(x₀), etc.)
• Substitute conditions after finding general solution
• Check units and physical meaning

Challenge 4: Verification Errors

Problem: Solution doesn’t satisfy original equation Solution:

• Double-check all derivative calculations
• Verify integration constants
• Use computer algebra systems for complex cases

Best Practices & Practical Tips

Before Solving

• Always classify the equation completely
• Check for obvious simplifications
• Consider physical meaning and expected behavior
• Identify all given conditions clearly

During Solution

• Show all steps clearly for verification
• Keep track of integration constants
• Use consistent notation throughout
• Check intermediate steps when possible

After Finding Solution

• Always verify by substitution
• Check that initial/boundary conditions are satisfied
• Analyze solution behavior (stability, growth, oscillation)
• Consider physical reasonableness

Common Mistakes to Avoid

• Forgetting absolute value signs in logarithms
• Losing or incorrectly applying constants of integration
• Mixing up homogeneous and particular solutions
• Not checking the domain of validity

Computational Tips

• Use graphing tools to visualize solutions
• Employ computer algebra systems for verification
• Consider numerical methods for complex equations
• Create solution templates for common types

Quick Reference Formulas

Integration Shortcuts

• ∫e^(ax)dx = (1/a)e^(ax) + C
• ∫sin(ax)dx = -(1/a)cos(ax) + C
• ∫cos(ax)dx = (1/a)sin(ax) + C
• ∫1/(ax+b)dx = (1/a)ln|ax+b| + C

Common Derivatives

• d/dx[e^(ax)] = ae^(ax)
• d/dx[sin(ax)] = a cos(ax)
• d/dx[cos(ax)] = -a sin(ax)
• d/dx[ln(ax)] = 1/x

Resources for Further Learning

Essential Textbooks

• “Elementary Differential Equations” by Boyce & DiPrima – Comprehensive introduction with applications
• “Differential Equations with Applications” by Zill – Strong on methods and examples
• “Introduction to Ordinary Differential Equations” by Ross – Clear explanations and problem sets

Online Resources

• Khan Academy – Free video tutorials on differential equations basics
• MIT OpenCourseWare – Complete differential equations course materials
• Wolfram Alpha – Equation solver and step-by-step solutions
• Paul’s Online Math Notes – Detailed notes with worked examples

Software Tools

• MATLAB/Octave – Numerical solutions and plotting
• Mathematica/Wolfram – Symbolic and numerical solving
• Python (SciPy) – odeint and solve_ivp functions
• Desmos Graphing Calculator – Visualization of solutions

Practice Resources

• Schaum’s Outline of Differential Equations – Large collection of solved problems
• Stewart’s Calculus – Good introduction with engineering applications
• Online problem generators – For additional practice problems

Advanced Topics to Explore Next

• Partial differential equations
• Systems of differential equations
• Numerical methods (Euler, Runge-Kutta)
• Laplace transforms
• Series solutions
• Stability analysis

This cheatsheet provides a foundation for solving basic differential equations. Master these fundamentals before advancing to more complex topics and applications.