ODEs vs PDEs

PDEs · Axiom Academy

Discover the fundamental difference between ordinary and partial differential equations through interactive visualizations! Step 1: Ordinary Differential Equations (ODEs) An ordinary differential equation involves a function of a single independent variable and its derivatives. Let's explore a classic example: exponential growth! Adjust the sliders to see how ODEs produce curves (1D solutions) Step 2: ODEs Create One-Dimensional Solutions Because an ODE has only one independent variable, its solution is a curve in 2D space . Let's see multiple solutions together to understand the solution space. Watch how different initial conditions create a family of curves One independent variable: time t Solution: A curve showing how y evolves over time Ordinary derivative: dy/dt - rate of change in one direction Step 3: Partial Differential Equations (PDEs) A partial differential equation involves a function of multiple independent variables and its partial derivatives. The classic example: the heat equation! The symbol ∂ (curly d) indicates a partial derivative . When we write ∂u/∂t, we mean "how does u change with respect to t while holding x fixed?" Similarly, ∂u/∂x means "how does u change with respect to x while holding t fixed?" Step 4: PDEs Create Multi-Dimensional Solutions Because a PDE has multiple independent variables, its solution is a surface (or higher-dimensional object). Let's visualize how heat diffuses along a rod over time.

This is the written version of the interactive lesson above. See the full PDEs course.