What is a Differential Equation? | Differential Equations Mastery

Welcome to Differential Equations!

The language of change in engineering and science

A differential equation is simply an equation that contains derivatives. Instead of solving for a number (like algebra), you solve for a function - the unknown is y(x), not just x.

Differential equations describe how things change. Newton's F = ma is really a differential equation: F = m(d²x/dt²). The motion of planets, the flow of current in circuits, the spread of diseases - all described by differential equations.

Differential Equation

Differential Equation (DE)

An equation containing an unknown function and one or more of its derivatives.

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The Recipe Analogy

Think of a differential equation as a recipe that tells you how something changes. If you know the recipe (the DE) and where you start (initial conditions), you can figure out the entire future behavior.

For example, dy/dt = 0.05y says 'the rate of change is 5% of the current value' - this describes bank interest or population growth!

Key Terminology

Classifying DEs

Medium

Classify: d²y/dx² + 3(dy/dx)² - y = sin(x)

1

Find the order

Highest derivative is d²y/dx² → Second-order

2

Check linearity

(dy/dx)² appears - that's y' squared → Nonlinear

3

Check type

Only one independent variable (x) → ODE

Answer: Second-order, nonlinear ODE

Quick Check

1. What is the order of: y''' + y' - y = 0

First-order

Second-order

Third-order

Zero-order

y''' means d³y/dx³, which is the third derivative. This is a third-order ODE.

2. Is y' + y² = x linear?

Yes

No

No - y² makes it nonlinear. In a linear DE, y and its derivatives can only appear to the first power.

Key Takeaways

A DE is an equation with derivatives - you solve for a function
Order = highest derivative present
Linear = y and derivatives appear to first power only
ODE = one independent variable, PDE = multiple