Differential Equation Fundamentals

Differential equations relate a function to its derivatives — the language of dynamic systems. Learn ODE vs PDE, order and degree, linear vs non-linear, initial and boundary conditions, general vs particular solutions, existence and uniqueness, direction fields, and the modelling workflow from physical law to equation.

Newton's second law, Kirchhoff's laws with capacitors and inductors, heat conduction, population growth and chemical kinetics all produce differential equations. Solving them predicts how systems evolve in time or vary in space.

An analogy

A differential equation is a rule for how something changes right now; solving it is like following the rule step by step to find where you'll be later.

Classification

ODE: one independent variable; PDE: several. Order = highest derivative. Linear: y and its derivatives appear to the first power, not multiplied together; otherwise non-linear (e.g. pendulum sin θ).

Solutions and conditions

The general solution contains n arbitrary constants for an n-th-order ODE; initial or boundary conditions fix them to give a particular solution. IVPs specify everything at one time; BVPs at two locations (beams, heat in rods).

Direction fields

For y′ = f(t, y), draw short slope marks at grid points; solution curves follow them. Equilibria (f = 0) and their stability can be read qualitatively.

Modelling workflow

Physical law (balance of mass, energy, momentum, charge) → assumptions → variables and parameters → ODE with conditions → solve (analytic or numeric) → validate against data → refine.

Formulas

• General linear ODE: aₙ(t)y⁽ⁿ⁾ + … + a₁(t)y′ + a₀(t)y = g(t)
• Newton's second law: m x″ + c x′ + k x = F(t)
• Series RC circuit: RC dv/dt + v = V_s(t)
• Existence & uniqueness: y′ = f(t, y): f and ∂f/∂y continuous ⇒ unique solution near (t₀, y₀)

Worked example: Classify

Given: (a) y″ + 3y′ + 2y = sin t (b) y′ = y² (c) ∂u/∂t = k ∂²u/∂x²

Find: Type, order, linearity.

• (a) ODE, 2nd order, linear
• (b) ODE, 1st order, non-linear (y²)
• (c) PDE, 2nd order in x, linear

Answer: As listed

Worked example: Verify a solution

Given: y = C e^(−2t) for y′ + 2y = 0, y(0) = 5

Find: Particular solution.

• y′ = −2Ce^(−2t) → y′ + 2y = 0 ✓
• y(0) = C = 5

Answer: y = 5e^(−2t)

Common mistakes

• Wrong: Too few conditions for the order. — Right: n-th order needs n conditions.
• Wrong: Calling y y′ = 1 linear. — Right: Product of y and y′ is non-linear.

Applications

• Circuits: RC, RL, RLC transients.
• Mechanics: Vibration, vehicle dynamics.
• Chemical & bio: Reaction kinetics, population models.

Key takeaways

• Classify: ODE/PDE, order, linearity.
• n-th order → n constants → n conditions.
• IVP vs BVP.
• Balance laws produce ODEs.

Going further

• Picard's theorem: Existence and uniqueness proof.
• Phase portraits: Qualitative analysis (M24).
• Stiff equations: Special numerical solvers (M9).