Differential Equations Basics

Differential Equations

Definition

• A differential equation is a mathematical equation that involves an unknown function and its derivatives.
• It relates the values of the function and its derivatives at different points in space and/or time.

Types of Differential Equations

• Ordinary Differential Equations (ODEs): involve a function of one independent variable and its derivatives.
• Partial Differential Equations (PDEs): involve a function of multiple independent variables and its partial derivatives.

Applications

• Physics and Engineering: model population growth, electrical circuits, and mechanical systems.
• Biology: model population dynamics, chemical reactions, and epidemiology.
• Economics: model economic systems, including supply and demand.

Français (Mathematics in French)

Vocabulary

• Équation différentielle: differential equation
• Fonction: function
• Dérivée: derivative
• Équation ordinaire: ordinary differential equation
• Équation aux dérivées partielles: partial differential equation

Famous French Mathematicians

• Pierre-Simon Laplace: developed the nebular hypothesis and worked on differential equations.
• Joseph-Louis Lagrange: made significant contributions to differential equations and calculus.

PCT (Percent Change and Trends)

Percent Change

• Formula: (new value - old value) / old value × 100
• Interpretation: the percentage increase or decrease from the original value.

Trends

• Increasing Trend: a pattern of increasing values over time.
• Decreasing Trend: a pattern of decreasing values over time.
• Stable Trend: a pattern of relatively constant values over time.

Applications

• Business: track changes in stock prices, sales, and revenue.
• Economics: analyze changes in GDP, inflation, and employment rates.
• Science: study changes in population growth, temperature, and other environmental factors.

Differential Equations

Definition

• Mathematical equations that link an unknown function with its derivatives.
• Describes relationships between a function's values and its derivatives across various dimensions like time and space.

Types of Differential Equations

• Ordinary Differential Equations (ODEs): Concern one independent variable and its derivatives.
• Partial Differential Equations (PDEs): Involve functions of multiple independent variables along with their partial derivatives.

Applications

• Physics and Engineering: Used to model phenomena such as population growth, electrical circuits, and mechanical systems.
• Biology: Applied in modeling population dynamics, chemical reactions, and the progression of diseases.
• Economics: Utilized in modeling economic behaviors, including aspects of supply and demand.

Français (Mathematics in French)

Vocabulary

• Équation différentielle: Differential equation
• Fonction: Function
• Dérivée: Derivative
• Équation ordinaire: Ordinary differential equation
• Équation aux dérivées partielles: Partial differential equation

Famous French Mathematicians

• Pierre-Simon Laplace: Noted for developing the nebular hypothesis and contributions to the study of differential equations.
• Joseph-Louis Lagrange: Renowned for his impactful work in differential equations and calculus.

PCT (Percent Change and Trends)

Percent Change

• Formula: (new value - old value) / old value × 100 which calculates the rate of change from one value to another.
• Interpretation: Represents the percentage rise or fall from an original value, indicating growth or decline.

Trends

• Increasing Trend: Characterized by a continuous rise in values over a duration.
• Decreasing Trend: Describes a consistent decline in values over time.
• Stable Trend: Exhibits values that remain relatively unchanged over a period.

Applications

• Business: Essential for tracking fluctuations in stock prices, sales figures, and overall revenue.
• Economics: Key to analyzing statistical changes in GDP, inflation rates, and employment levels.
• Science: Important in examining shifts in variables like population dynamics, temperature changes, and environmental factors.

Differential Equations

• A differential equation involves an unknown function and its derivatives, showing the relationship between them across various domains such as physics and biology.
• Types of differential equations include:
  • Ordinary Differential Equations (ODEs): Function of one independent variable with derivatives (e.g., dy/dx = f(x,y)).
  • Partial Differential Equations (PDEs): Function of multiple independent variables with partial derivatives (e.g., ∂u/∂t = ∂²u/∂x²).

Methods for Solving Differential Equations

• Direct Integration: Solving by integrating both sides of the equation.
• Separation of Variables: Isolating variables and integrating each side.
• Integrating Factor: Introducing a factor to make the left side of the equation exact for easier integration.

Français (Mathematics in French)

• Les mathématiques translates to "mathematics" in French.
• Équation différentielle refers to "differential equation."
• Dérivé means "derivative."
• Fonction translates to "function."

PCT (Percent Change and Trends)

• Percent Change measures quantity alterations over time as a percentage, calculated by the formula: ((New Value - Old Value) / Old Value) × 100.
• Trends illustrate patterns or directions of change over time; types include:
  • Increasing Trend: Values rise over time.
  • Decreasing Trend: Values fall over time.
  • Stable Trend: Values remain roughly constant.

Analyzing Trends

• Identifying the trend involves determining its direction and strength.
• Calculating Percent Change measures variations between values.
• Making predictions utilizes trends to forecast future values.