Partial Differential Equations

What Are Partial Differential Equations?

A partial differential equation (PDE) is simply an equation that involves:

An unknown function (like )
The partial derivatives of that function (like or )

Think of it this way: if you have a function that depends on multiple variables (like temperature depending on both position and time), a PDE describes how that function changes.

Simple Examples:

(the function plus its derivatives equals something)
(the function multiplied by its derivative)

Key Properties of PDEs

Order

The order is simply the highest derivative in the equation.

First-order: involves ,
Second-order: involves , , etc.

Degree

The degree is the highest power of the highest-order derivative (after cleaning up fractions and square roots).

Linear vs Non-linear

Linear PDE: The unknown function and its derivatives appear “nicely” - no powers, no products between them. Non-linear PDE: Has powers of derivatives or products between derivatives.

Helpful Notation (Shorthand)

Instead of writing long partial derivatives, mathematicians use shortcuts:

For a function :

(how changes with )
(how changes with )

For second derivatives:

(mixed derivative)

Types of First-Order PDEs (From Simple to Complex)

1. Linear PDEs

What they look like:

Key feature: Everything is “straight” - no powers, no products between , , and .

Examples:

(very simple!)

2. Semi-linear PDEs

What they look like:

Key feature: The derivatives and are linear, but the right side can involve in complicated ways.

Examples:

(notice on the right)

3. Quasi-linear PDEs

What they look like:

Key feature: The coefficients of and can depend on too, but and themselves are still linear.

Examples:

4. Non-linear PDEs

What they look like: Anything that doesn’t fit the above patterns.

Key feature: Has powers of derivatives or products between derivatives.

Examples:

(has and )
(product of derivatives)

Lagrange’s Method: Solving Quasi-linear PDEs

The Problem

We want to solve equations of the form:

where , , and are functions of , , and .

The Big Idea

Instead of working directly with the PDE, we convert it to a system of ordinary differential equations (which are easier to solve).

The Magic Formula

For the PDE , we create the auxiliary equations:

Think of this as three ratios that are all equal.

Step-by-Step Solution Method

Step 1: Put your PDE in the form

Step 2: Write the auxiliary equations:

Step 3: Solve these equations to find two independent solutions:

(first constant)
(second constant)

Step 4: The general solution is any relationship between and : where is any function you choose.

Important Tips

You need two independent solutions (meaning one isn’t just a multiple of the other)
At least one solution must involve
The auxiliary equations are often solved by pairing them up (like )

Why This Works (Intuitive Explanation)

The auxiliary equations represent characteristic curves in 3D space. Along these curves, the original PDE becomes much simpler to handle. The solutions and are like “coordinates” that remain constant along these special curves.

Practice Problems to Try

Work through these examples to build your understanding:

Easy:
Medium:
Simple:
Trigonometric:
Basic: