Differential Equations Formulas | First Order, Second Order & Partial Equations

First Order Ordinary Differential Equations

Definition: These equations involve an unknown function and its first derivative.

Conceptual Meaning: They describe how one quantity changes with respect to another, capturing the idea of rates of change.

Geometric Interpretation: Represented visually using slope fields that indicate the direction of change at every point.

Real-Life Applications:

• Population Growth and Decay: Modeling changes in populations over time.
• Cooling and Heating: Newton’s law of cooling.
• Motion Problems: Relating velocity and displacement.

Second Order Ordinary Differential Equations

Definition: These equations involve second derivatives, often describing acceleration or curvature.

Conceptual Understanding: Solutions depend not only on the equation but also on initial conditions.

Applications:

• Simple Harmonic Motion: Springs, pendulums, and oscillations.
• Electrical Circuits: Behavior of LCR (inductor-capacitor-resistor) circuits.
• Vibrations: Used in sound, mechanics, and material testing.

Differential Calculus Formulas | Functions, Limits, Derivatives & Applications

Some Partial Differential Equations

Definition: Equations involving partial derivatives with respect to more than one variable.

Meaning: They model systems where change occurs across both space and time.

Physical Interpretation: Useful for describing heat, waves, or fluids that evolve dynamically.

Applications:

• Heat Conduction: Heat equation models how temperature spreads.
• Wave Propagation: Wave equation models sound, light, and water waves.
• Fluid Flow and Diffusion: Models chemical reactions and physical processes.

Integral Calculus Formulas | Indefinite, Definite, Double & Triple Integrals