Differential Equations Toolkit (IPhO)

Most IPhO differential-equation work is pattern recognition: spot the template, do the one substitution that simplifies it, then use initial conditions to pin down constants.

1. Definitions (Must Know)

A. ODE, order, and notation

An ordinary differential equation (ODE) relates a function to its derivatives with respect to a single variable.

• Order 1: y' = dy/dx (or y dot = dy/dt).
• Order 2: y'' = d²y/dx² (or y double dot = d²y/dt²).

An ODE is linear in y if it can be written as

where y and its derivatives appear only to the first power and are not multiplied together.

B. General vs particular solution

• General solution: contains integration constants (C₁,C₂,…).
• Particular solution: one specific solution that satisfies a given driving/forcing term and/or initial conditions.

For linear ODEs, the full answer is typically:

C. Equilibrium (steady-state) value and time constant

Many physics ODEs relax to an equilibrium value y_∞:

• τ is the time constant (sets how fast you approach equilibrium).
• y_∞ is found by setting y dot = 0 (the long-time steady state).

D. Canonical “high-yield” forms

• Separable: dy/dx = f(x)g(y).
• First-order linear: dy/dx + P(x)y = Q(x).
• SHM: x double dot + ω² x = 0.
• Damped oscillator: x double dot + 2β x dot + ω₀² x = 0.

2. Key Ideas (What Earns Marks)

• Put the ODE into a standard template (separable, linear, constant-coefficient oscillator) before doing algebra.
• If the equation contains (y-y_∞), shift variables: let u = y-y_∞ so the steady-state becomes u → 0.
• Keep the integration constant(s) until after you apply the initial condition(s).
• For oscillators: classify the regime fast (underdamped vs overdamped) before writing a solution form.
• State your physics reasoning for the sign and the long-time limit (this often earns method marks):
  • t → ∞ should give y → y_∞ for a stable relaxation.
  • Damping should make amplitudes decay, not grow.
• Sanity checks: units, limiting cases (e.g. “turn off” drag/damping), and a quick derivative-substitution check.

3. Detailed Explanations

A. Template recognition table

B. Separable ODEs (the fastest win)

If

then

Two practical points:

1. Keep the constant explicit: write + C and apply the initial condition early.
2. Log constants combine: ln |y| + C = ln(C'|y|), but only after you have handled sign and initial conditions cleanly.

C. First-order linear ODEs (integrating factor)

Standard form:

Define the integrating factor

Then the left-hand side becomes an exact derivative:

So the solution is

D. Relaxation to equilibrium (cooling, RC/RL, terminal speed)

Many problems are “one step away” from the canonical relaxation form:

Solution:

How to find y_∞ quickly:

• Set y dot = 0 and solve the remaining algebraic equation.
• Interpret it physically as the long-time steady state (no more change).

Examples (same math, different symbols):

• Newton cooling: y = T, y_∞ = Tₐ, τ = 1/k.
• RC charging: y = V_C, y_∞ = V₀, τ = RC.
• Linear drag: y = v, y_∞ = vₜ, τ = m/b.

E. Second-order constant-coefficient ODEs (oscillators)

The most common physics form is

Try x = e^rt, giving the characteristic equation

Regimes:

• Underdamped ( β < ω₀): oscillatory decay
  x(t) = Ae^(-β t) cos(ω_d t + φ), ω_d = square root of (ω₀²-β²) .
• Critically damped ( β = ω₀): fastest non-oscillatory return
  x(t) = (A + Bt)e^(-β t).
• Overdamped ( β > ω₀): sum of two decays
  x(t) = C₁e^(r₁ t) + C₂e^(r₂ t), r_(1,2) = -β± square root of (β²-ω₀²) .

If the ODE is written as mx double dot + bx dot + kx = 0, then

F. Forcing: constant forces and sinusoidal drives

If the equation is

you almost never want to integrate from scratch. Two high-yield cases:

1. Constant forcing: if f(t) = a (constant), try a constant particular solution xₚ = a/ω₀² (for β = 0). More generally, find x_∞ by setting x dot = x double dot = 0 and solving ω₀² x_∞ = a. Then solve the homogeneous equation for u = x-x_∞.

2. Sinusoidal forcing: if f(t) = F₀ cos(ω t), guess a sinusoidal particular solution. In timed settings, you often only need:

   • the existence of a transient (homogeneous) part that decays with e^(-β t),
   • a steady-state oscillation at the driving frequency ω,
   • resonance intuition: response is largest near ω ≈ ω₀ when damping is small.

G. Small oscillations (linearise a nonlinear system)

If a particle moves in a potential U(x) and x₀ is a stable equilibrium, then

Taylor expand near x₀:

Then the force is approximately Hooke-like:

so

which is SHM with

H. Fast verification (worth doing under pressure)

• Differentiate your proposed solution once and substitute back (a quick check).
• Check t → ∞ (does it approach the right equilibrium?).
• Check “turn off” parameters: if b → 0, does damped motion reduce to SHM?
• Units: τ must have units of time; ω must have units of s⁻¹.

4. Common Mistakes

• Dropping integration constants too early (then you cannot satisfy initial conditions).
• Forgetting the shift u = y-y_∞ (so the particular solution is harder than it needs to be).
• Mixing up yₕₒₘ and yₚₐᵣₜ in linear ODEs (the answer is their sum).
• Writing the underdamped form when the parameters correspond to overdamping (check β vs ω₀ first).
• Losing a sign when separating variables (especially when moving (y-y_∞) across the equals sign).
• Solving for τ but forgetting it must be positive for a stable decay.

5. Exam Tips

• Show the classification explicitly: write “separable”, “linear with integrating factor”, or “constant-coefficient oscillator”.
• If the question asks only for a time constant or terminal value, you may not need the full solution:
  • terminal value: set derivatives to 0,
  • time constant: compare with e^(-t/τ).
• When applying initial conditions, do it at a convenient time (often t = 0) to reduce algebra.
• If you get exponentials growing in a damping/drag problem, stop and re-check the sign.

6. Worked Examples

A. Newton cooling / RC-style relaxation

Solve dT/dt = -k(T-Tₐ) with T(0) = T₀.

B. Terminal speed with linear drag

A particle falls with mdv/dt = mg-bv and v(0) = 0. Find v(t) and vₜ.

C. SHM with initial conditions

Solve x double dot + ω² x = 0 with x(0) = x₀ and x dot (0) = v₀.

D. Constant forcing on an oscillator (shift the equilibrium)

Solve x double dot + ω² x = a with x(0) = 0 and x dot (0) = 0.

7. Mind Stretchers

A. Quadratic drag (separable, but with a trick)

Solve mdv/dt = mg-cv² with v(0) = 0.

B. A linear ODE where the “right” variable is hidden

A quantity y obeys y dot = k(a-by) with constants a,b,k > 0. Find y(t) given y(0) = 0.

C. Damping regime without solving for constants

For mx double dot + bx dot + kx = 0, state the condition for oscillations, in terms of m,b,k.

8. Practice