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Ordinary Differential Equations

★★★☆☆High School

📖Definition

An ordinary differential equation (ODE) relates an unknown function of one variable to its derivatives. It's the fundamental language of physics and engineering.

📐Formulas

(dy)/(dx) = f(x, y)

General form of first-order ODE

y'' + p(x)y' + q(x)y = g(x)

Second-order linear ODE

(dⁿ y)/(dxⁿ) = F(x, y, y', ..., y^(n-1))

nth-order ODE

✏️Examples

예제 1

Solve dy/dx = 2x.

예제 2

Solve y' + y = 0.

📜History

Discovered by: Isaac Newton, Leibniz (17th century)

Developed alongside calculus to mathematically express physical laws.

⚡Applications

Physics

Newtonian mechanics, electromagnetism

Engineering

Circuits, control systems

Biology

Population models, epidemiology

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