1. What Is a Differential Equation?
Differential Equation (DE): An equation involving an unknown function and one or more of its derivatives.
ODE vs PDE
| Type | Variables | Derivatives | Example |
|---|
| ODE | One independent variable | Ordinary (d/dx) | y'' + 3y = 0 |
| PDE | Two or more independent variables | Partial (∂/∂x, ∂/∂t) | u_t = k u_xx (heat eqn) |
Order
Order = highest derivative appearing in the equation.
y''' + 2y' = x is third order. (y')³ + y = 0 is first order (power ≠ order).
Linearity
Linear ODE: a_n(x)y^(n) + ... + a_1(x)y' + a_0(x)y = g(x). The unknown y and all its derivatives appear to the first power only, with no products of y and its derivatives.
| Equation | Linear? | Why? |
|---|
| y'' + 5y = sin x | Yes | y terms to first power |
| y' + y² = 0 | No | y² term |
| yy' = x | No | Product of y and y' |
Solutions
General solution: Family containing n arbitrary constants for an n-th order ODE.
Particular solution: One specific member (constants determined by initial/boundary conditions).
IVP: DE + initial conditions: y(x_0)=y_0 (and derivatives for higher order).
To verify a solution: substitute it and its derivatives into the DE and confirm identity.
2. Direction Fields
Direction field: At each point (x,y), draw a segment with slope f(x,y). Solution curves follow these slopes.
Isocline: Curve where all slopes equal a constant c. Found by setting f(x,y) = c.
Autonomous Equations
dy/dx = f(y) -- right side depends only on y. Direction field has identical columns.
Equilibria
Equilibrium: constant solution y = c where f(c) = 0. Stable if nearby solutions approach; unstable if they flee.
3. Existence and Uniqueness
Picard-Lindelof Theorem: If f(x,y) and df/dy are continuous in a rectangle around (x_0, y_0), then the IVP dy/dx = f(x,y), y(x_0) = y_0 has a unique solution on some interval containing x_0.
| Condition | Guarantees |
|---|
| f continuous | Existence |
| f and df/dy continuous | Existence AND Uniqueness |
Classic failure: dy/dx = y^(1/3), y(0)=0. df/dy = (1/3)y^(-2/3) is undefined at y=0. Multiple solutions exist: y=0 and y=(2x/3)^(3/2).
4. Euler's Method
x_(n+1) = x_n + h
y_(n+1) = y_n + h * f(x_n, y_n)
Geometric idea: follow the tangent line for distance h, then recompute the slope. Global error is O(h).
Algorithm Steps
- Start at (x_0, y_0).
- Compute slope: m = f(x_n, y_n).
- Step forward: x_(n+1) = x_n + h, y_(n+1) = y_n + h*m.
- Repeat until reaching target x.
Smaller h = more steps = better accuracy (but more computation). Halving h roughly halves the global error.