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Axioma

Mathematics · Advanced · 35 min · Chapter 13 of 28

Differential equations

An equation relating a function to its derivatives. Separable equations, first-order linear equations, and the harmonic oscillator.

Read first: Integral calculus

The idea

A differential equation constrains the way a quantity can change. The order is the highest derivative that appears. An ordinary differential equation has one independent variable; a partial differential equation has several. A general solution of a first-order equation contains one arbitrary constant; initial conditions pin the constant down. Existence is not automatic from writing the equation, but for y′ = f(t, y) with f reasonably smooth, Picard’s theorem supplies a unique solution through a given point.

If the equation has the form dy/dx = g(x) h(y), separate: dy/h(y) = g(x) dx, and integrate both sides. You must watch where h(y) = 0; those constant solutions can be lost in the division. A first-order linear equation y′ + p(t) y = q(t) looks harder and is mechanical: multiply by the integrating factor μ = exp(∫ p dt). The left side becomes the derivative of μ y, and one more integration finishes the job.

Second-order linear equations with constant coefficients are solved by trying e^{rt} and solving the resulting polynomial. For x″ + ω² x = 0 the characteristic roots are ±iω, and the real solutions are A cos(ωt) + B sin(ωt). That is simple harmonic motion: oscillation at angular frequency ω, forever, because no damping was put in the equation. Add a first-derivative term and the motion decays.

Hold onto this

  1. The unknown is a function, not a number. A solution is any function that satisfies the equation on an interval.
  2. A first-order equation plus a point determines a unique solution, under a Lipschitz condition.
  3. Separable equations yield to algebra and one integral. Linear equations yield to an integrating factor.
  4. The oscillator x″ + ω²x = 0 is the source of sine and cosine in physics.

Formulas

Separable equation

Divide by h(y), multiply by dx, and integrate both sides. Check constant solutions h(y) = 0 separately.

g, h
Functions of one variable each

Integrating factor

For y′ + p(t) y = q(t), multiply through by μ. Then (μ y)′ = μ q.

p(t)
The coefficient of y

Simple harmonic oscillator

Acceleration is proportional to displacement, toward the origin. Solutions oscillate at angular frequency ω.

ω
Angular frequency, in radians per time

Oscillator solution

The general solution. A and B are fixed by two initial conditions, position and velocity.

A, B
Constants

Logistic equation

Growth rate r when scarce, shutting off as N approaches the carrying capacity K.

K
Carrying capacity
r
Low-density growth rate

Worked examples

1. Separate and solve

Solve dy/dx = 2x / y, with y(0) = 3. Take the positive branch.

  1. y dy = 2x dx.
  2. Integrate: y²/2 = x² + C.
  3. y(0) = 3 gives 9/2 = C, so y² = 2x² + 9.
  4. y = √(2x² + 9), the positive root, since y(0) is positive.

y = √(2x² + 9).

2. A spring

Solve x″ + 4x = 0 with x(0) = 1 and x′(0) = 0.

  1. Here ω² = 4, so ω = 2.
  2. x(t) = A cos(2t) + B sin(2t).
  3. x(0) = A = 1.
  4. x′(t) = −2A sin(2t) + 2B cos(2t), so x′(0) = 2B = 0. Thus B = 0.

x(t) = cos(2t). It starts at 1 with zero velocity and oscillates with period π.

Try these

Find the integrating factor for y′ + 3y = eˣ.

What constant solutions does the logistic equation have?

Common mistakes

  • Dividing by y, or by h(y), silently drops the solution y = 0 when that solution exists.
  • The general solution of a second-order equation needs two constants. One initial condition does not determine it.
  • ω in the oscillator is angular frequency, not the frequency in cycles. The period is 2π/ω.