cheat sheet for differential equations

Guri Bee

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22-10-2024

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Differential Equations: A Quick Reference Guide

Differential equations are the backbone of many scientific and engineering disciplines. They describe the relationship between a function and its derivatives, often modeling real-world phenomena like population growth, heat transfer, or the motion of objects. This cheat sheet provides a concise overview of key concepts and methods for solving differential equations.

Types of Differential Equations

1. Ordinary Differential Equations (ODEs): Involve only one independent variable.

• Example: The equation dy/dt = ky models exponential growth, where y is the population and t is time.

2. Partial Differential Equations (PDEs): Involve multiple independent variables.

• Example: The heat equation ∂u/∂t = k(∂²u/∂x² + ∂²u/∂y²) describes how temperature u changes over time t and spatial coordinates x and y.

Order and Linearity

Order: Determined by the highest derivative present.

• First-order: dy/dx + y = 0
• Second-order: d²y/dx² + 3dy/dx + 2y = 0

Linearity: A differential equation is linear if the dependent variable and its derivatives appear only to the first power and are not multiplied together.

• Linear: dy/dx + y = sin(x)
• Nonlinear: dy/dx = y²

Common Solution Techniques

1. Separation of Variables: Suitable for first-order ODEs where the variables can be separated.

• Example: Solving the equation dy/dx = xy involves separating the terms to get dy/y = x dx and integrating both sides.

2. Integrating Factors: Used to solve linear first-order ODEs of the form dy/dx + p(x)y = q(x).

• Example: The equation dy/dx + 2y = x can be solved by multiplying both sides by the integrating factor e^(2x).

3. Homogeneous Equations: A special type of first-order ODE where the right-hand side can be written as a function of y/x.

• Example: The equation dy/dx = (x² + y²)/xy is homogeneous because the right-hand side can be expressed as (1 + (y/x)²)/(y/x).

4. Exact Equations: A special type of first-order ODE that can be written as the total differential of some function.

• Example: The equation M(x, y) dx + N(x, y) dy = 0 is exact if ∂M/∂y = ∂N/∂x.

5. Linear Second-Order Homogeneous Equations: Can be solved using the characteristic equation, which is a quadratic equation whose roots determine the form of the solution.

• Example: The equation d²y/dx² + 3dy/dx + 2y = 0 has the characteristic equation r² + 3r + 2 = 0, whose roots are r = -1 and r = -2. The general solution is then y(x) = c₁e^(-x) + c₂e^(-2x).

Additional Notes:

• Initial Conditions: Differential equations often require initial conditions to determine the unique solution. These conditions specify the value of the function or its derivative at a specific point.

• Numerical Methods: For complex or non-analytical problems, numerical methods like Euler's method, Runge-Kutta methods, and finite difference methods can be used to approximate solutions.

• Applications: Differential equations are widely used in various fields, including:

  • Physics: Modeling motion, forces, and energy.
  • Biology: Describing population growth, disease spread, and drug kinetics.
  • Finance: Analyzing stock prices and financial models.
  • Engineering: Designing structures, analyzing circuits, and controlling systems.

Resources:

• Khan Academy: Differential Equations
• Paul's Online Notes: Differential Equations

Remember: This cheat sheet serves as a quick reference for key concepts and techniques. It's essential to delve into the specific methods and practice solving various differential equations to gain mastery.