x'' = -ω²x
Equations of Motion
Differential Equations & The Law of Change
The Central Question
In World 18, we calculated how fast something changes. But Physics often gives us the law governing the change and asks: What is the motion? An ordinary equation tells us what a quantity must equal. A differential equation tells us how a quantity must evolve. We must find the function that obeys the law.
QUANTITY →
FUNCTION →
RATE OF CHANGE →
DERIVATIVE →
LAW OF CHANGE →
DIFFERENTIAL EQUATION →
SOLUTION →
FAMILY OF MOTIONS →
INITIAL CONDITIONS →
SPECIFIC MOTION
The 59 Motion Cells
From Laws of Change to Dynamic Evolution
CELL 01-07
Foundations
dy/dx= f(x)
→
- Function as the Unknown
- Algebraic vs Differential Eq
- General Solution (+C)
- Particular Solution
- Initial & Boundary Conditions
CELL 08-13
Classification
(y'')3 + y' = x
- Order (Highest Derivative)
- Degree (Power of Highest)
- Order vs Degree Distinction
- Linear vs Nonlinear ODEs
CELL 14-21
First-Order Equations
dy/y
x dx
- Separation of Variables
- Linear First-Order (dy/dx + Py = Q)
- Integrating Factor (e^∫Pdx)
- Homogeneous Equations
- Exact Equations (Preview)
CELL 22-27
Model Equations
- Autonomous Equations (dy/dt = f(y))
- Exponential Growth & Decay
- Logistic Growth (Carrying Capacity)
- Equilibrium Solutions
CELL 28-34
Motion
- Second-Order Equations (d²x/dt²)
- Constant Acceleration Derived
- Newton's Second Law as an ODE
- Simple Harmonic Motion (x'' = -ω²x)
CELL 35-41
Applications
- Mixing Problems (Rate in - Rate out)
- Newton's Law of Cooling
- Reaction Rates (Chemistry)
- Economic Growth (Capital)
CELL 42-46
Higher Equations
- Higher-Order Linear ODEs
- Characteristic Equation (ar² + br + c = 0)
- Repeated & Complex Roots
- Numerical Solutions (Euler/RK)
CELL 47-49
PDE Frontier
- ODE vs PDE (Multiple Variables)
- The Wave Equation (u_tt = c²u_xx)
- Partial Differential Equations Preview
CELL 50-53
Dynamical Systems
- Existence & Uniqueness
- Dynamical Systems Behavior
- Phase Space Trajectories
- Stability (Stable/Unstable Equilibria)
CELL 54-59
The Change Engine Synthesis
Law
Diff Eq
Solution
- Diff Eq as Law of Change
- Local Law → Global Trajectory
- The Calculus Loop (d/dx ↔ ∫)
- School Kinematics Reinterpreted
Exit Portal
Mathematics stops merely describing static objects and begins describing entire worlds evolving through time. The next major door is not simply "more calculus." A one-dimensional motion equation has now prepared us to ask: What happens when motion, force, change, and geometry all exist in two or three dimensions?