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Mathematics for Physics

15 Differential Equations

Differential equations is the study of equations that relate an unknown function to its derivatives.

A derivative is a rate: the rate of change of one variable with respect to another. An unknown is a function: the quantity whose formula is found by solving the equation.

Differential equations is related to calculus because the law relating a quantity to its rate of change is independent of how the unknown function is named and of which coordinates describe the same change.

Here are the ideas that are fundamental to study of differential equations.

Existence and uniqueness of solutions. Given a starting value and a continuity requirement, a differential equation has exactly one solution. A continuity requirement is a smoothness condition. A solution is a function that satisfies the equation.

Superposition for linear equations. For a linear differential equation, a sum of solutions scaled by constants is again a solution. A linear differential equation is an equation in which the unknown and its derivatives appear to the first power.

Initial and boundary conditions. A differential equation has a family of solutions. Initial conditions and boundary conditions select one solution. An initial condition is a starting value. A boundary condition is a value at an edge of the domain.

15.2 References

  1. Simmons, G. F. Differential Equations with Applications and Historical Notes. — existence and uniqueness under a continuity requirement.
  2. Farlow, S. J. An Introduction to Differential Equations and Their Applications. — superposition for linear equations, and applications to growth, heat, and waves.
  3. Stewart, J. Calculus: Early Transcendentals. — a differential equation as a relation among a function and its derivatives.
  4. Logan, J. D. A First Course in Differential Equations. — the unknown as a function, and initial and boundary conditions.