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Integrating FactorEdit

Integrating factor is a standard technique in the toolkit of methods for solving certain differential equations. In its classic form, it is used to solve linear first-order differential equations of the type y′(x) + p(x) y(x) = q(x). The essence of the method is to multiply the entire equation by a carefully chosen function μ(x) so that the left-hand side becomes the derivative of a product, enabling straightforward integration. This approach not only yields explicit solutions but also highlights the deeper structure of linear systems, where changes of variables reveal hidden simplicity.

The integrating factor method rests on a simple idea: convert a non-separable equation into a form that is easy to integrate by recognizing when a product rule can be invoked. The key step is selecting μ(x) so that (μ(x) y(x))′ equals μ(x) q(x). When μ is chosen as exp(∫ p(x) dx), the left-hand side becomes the exact derivative of μ y, and the equation can be integrated directly.

Derivation and standard form

  • The starting point is the linear first-order differential equation in the standard form y′ + p(x) y = q(x), with p and q continuous on an interval I.
  • The integrating factor μ(x) is defined by μ(x) = exp(∫ p(x) dx). This is always positive and never zero on any interval where p is integrable.
  • Multiplying the entire equation by μ gives μ y′ + μ p y = μ q.
  • By the product rule, μ y′ + μ p y = (μ y)′. Thus (μ y)′ = μ q.
  • Integrating both sides with respect to x yields μ(x) y(x) = ∫ μ(x) q(x) dx + C, where C is a constant of integration.
  • Solving for y gives the general solution y(x) = [∫ μ(x) q(x) dx + C] / μ(x). An equivalent way to express this is y(x) = e^(-∫ p(x) dx) [∫ e^(∫ p(x) dx) q(x) dx + C].

This approach makes the method transparent: if you can identify p(x) and q(x), you can construct μ(x) and reduce the problem to a direct integral.

Worked examples

  • Example 1: Solve y′ + 2y = e^(2x).

    • p(x) = 2, q(x) = e^(2x). The integrating factor is μ(x) = exp(∫ 2 dx) = e^(2x).
    • Multiply: e^(2x) y′ + 2 e^(2x) y = e^(2x) e^(2x) ⇒ (e^(2x) y)′ = e^(4x).
    • Integrate: e^(2x) y = ∫ e^(4x) dx = (1/4) e^(4x) + C.
    • Solve for y: y(x) = e^(-2x) [(1/4) e^(4x) + C] = (1/4) e^(2x) + C e^(-2x).

  • Example 2: Solve y′ + (1/x) y = sin x / x for x > 0.

    • p(x) = 1/x, q(x) = sin x / x. The integrating factor is μ(x) = exp(∫ (1/x) dx) = x.
    • Multiply: x y′ + y = sin x.
    • Left side is (x y)′, so integrate: (x y)′ = sin x ⇒ x y = -cos x + C.
    • Solve for y: y(x) = (-cos x + C) / x.

Special cases and remarks

  • If q(x) ≡ 0, the equation is homogeneous and the solution reduces to y(x) = C e^(-∫ p(x) dx).
  • The method presumes p and q are well-behaved (continuous) on an interval; under standard existence-uniqueness results, an initial value problem y(x0) = y0 has a unique solution on the interval where these conditions hold.
  • The integrating factor technique is also a gateway to connecting linear equations with the concept of an exact differential. In more general first-order forms M(x, y) dx + N(x, y) dy = 0, an integrating factor μ can render the differential exact, transforming the problem into a potential function whose differential matches μ M dx + μ N dy.

Connections and extensions

  • This method is a cornerstone of the theory of Differential equations and is taught as part of the study of First-order linear differential equations and Linear differential equations.
  • The product-rule perspective behind integrating factors links to the idea of an Exact differential: multiplying by μ can convert a non-exact equation into an exact one, enabling straightforward integration.
  • Beyond scalar equations, integrating factors appear in more advanced settings such as systems of linear differential equations and certain partial differential equations, where they help decouple or simplify the system.
  • In physics and engineering, integrating factors underlie the analytic solution of many time-dependent processes, including RC and RL circuits, heat conduction problems, and population dynamics modeled by linear response terms. See, for example, the use of integrating factors in RC circuit analysis and in introductory treatments of Differential equations in physics.

See the broader context of these ideas in the study of Differential equation, First-order linear differential equation, and Exponential function.

See also