Back to Module 1

Module 1 Quick Reference

Introduction to Differential Equations

Classification

ODE vs PDE

ODE: one independent variable (ordinary derivatives)

PDE: two+ independent variables (partial derivatives)

Order

= highest derivative present

Note: (y')^3 is order 1, not 3

Linear

y, y', y'',... appear to 1st power only; no products of y terms

Coefficients may depend on x

Solutions

General Solution

n-th order ODE has n arbitrary constants

Particular Solution

Constants fixed by initial conditions

Verification

Substitute y and its derivatives back into the DE; check for identity

Direction Fields & Equilibria

Direction Field

Plot slope f(x,y) at grid points

Solution curves follow the flow

Isocline for slope c: f(x,y) = c

Autonomous: dy/dx = f(y)

Slopes depend only on y (identical columns)

Equilibria: f(c) = 0 gives y = c

Stable: nearby solutions approach

Unstable: nearby solutions flee

Existence & Uniqueness (Picard-Lindelof)

If f and df/dy are continuous near (x_0, y_0), then the IVP has a UNIQUE local solution.
f continuousf and df/dy continuous
Existence guaranteedExistence AND Uniqueness guaranteed

Classic failure: dy/dx = y^(1/3), y(0)=0. df/dy undefined at y=0. Multiple solutions.

Euler's Method

x_(n+1) = x_n + h     y_(n+1) = y_n + h * f(x_n, y_n)

Algorithm

  • Start at (x_0, y_0)
  • Compute slope m = f(x_n, y_n)
  • Step: y += h*m, x += h
  • Repeat

Error

Local: O(h^2) per step

Global: O(h) over interval

Halving h halves the error

Smaller h = better accuracy