Exact differential equation

An exact differential equation is a first-order ordinary differential equation of the form M(x,y) \, dx + N(x,y) \, dy = 0, where M and N are continuous functions such that there exists a function \psi(x,y) satisfying \frac{\partial \psi}{\partial x} = M and \frac{\partial \psi}{\partial y} = N, making the left-hand side the total differential d\psi = 0.^[1]^[2] This property ensures that the equation is path-independent in integration, allowing solutions to be obtained directly through algebraic integration without additional techniques like integrating factors.^[3] The key test for exactness is the equality of mixed partial derivatives: \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}, which follows from the continuity of the second partial derivatives of \psi by Clairaut's theorem.^[1]^[2] If this condition holds, the solution is given implicitly by \psi(x,y) = C, where C is a constant, found by integrating M with respect to x (treating y as constant) and adding a function of y determined by the N term.^[3] Exact equations arise naturally in contexts where the differential represents a conservative field or an exact change in a state function, such as in thermodynamics for properties like internal energy or entropy.^[3] Exact equations form a distinct class of first-order differential equations that overlaps with separable and linear ones; their significance lies in their direct solvability and broader implications in multivariable calculus and physics, where non-exact forms may require adjustments to achieve exactness.^[1] Higher-order exact equations can be reduced to first-order by substitution, but the core concept remains tied to the existence of a potential function.^[3]

Fundamentals

Definition

In multivariable calculus, the total differential of a function F(x, y) represents the infinitesimal change in F and is expressed as

dF = \frac{\partial F}{\partial x}\, dx + \frac{\partial F}{\partial y}\, dy.

This form arises from the chain rule applied to the function's partial derivatives, capturing how small changes in the independent variables x and y affect F.^[1] An ordinary differential equation (ODE) of the first order is termed exact if it can be expressed in the form dF(x, y) = 0 for some function F(x, y), where the level curves F(x, y) = c (with c constant) provide the general solution to the equation.^[1] The standard general form of such an ODE is P(x, y)\, dx + Q(x, y)\, dy = 0, which is exact if there exists a function F(x, y) such that P = \frac{\partial F}{\partial x} and Q = \frac{\partial F}{\partial y}.^[1] This notion of exactness draws motivation from thermodynamics, where exact differentials correspond to changes in state functions (such as internal energy, whose value depends only on the system's state), in contrast to inexact differentials associated with path-dependent quantities (such as heat or work, which vary with the process taken).^[4]^[5]

Historical Context

The concept of exact differential equations originated in the 18th century with Leonhard Euler's foundational work on first-order ordinary differential equations. In his 1763 paper De integratione aequationum differentialium, Euler introduced the idea of integrating factors to render non-exact equations integrable, presenting a theorem that any equation of the form M \, dx + N \, dy = 0 becomes exact upon multiplication by a suitable factor L(x, y).^[6] He expanded this framework in Institutionum Calculi Integralis (1768–1770), where he systematically treated exact equations as total differentials and provided methods for their integration, emphasizing generality in solving P \, dx + Q \, dy = 0 when \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}.^[7] Joseph-Louis Lagrange built upon Euler's contributions in the late 18th and early 19th centuries, developing the calculus of variations and applying it to mechanics in works such as Mécanique Analytique (1788). These efforts involved principles of path independence in optimization, related to concepts underlying exact differentials.^[8] The 19th century saw significant generalizations through Pfaffian differentials and higher-order extensions. Johann Friedrich Pfaff advanced the theory around 1814–1815 by studying systems of partial differential equations of the first order, introducing Pfaffian forms to characterize integrability conditions for exact differentials in multiple variables.^[9] Carl Gustav Jacob Jacobi further developed these ideas in the 1830s–1840s, generalizing to higher dimensions and applying them to elliptic integrals and determinants, which facilitated solutions for multivariable exact systems. Applications in physics emerged prominently in the 1800s, particularly in the context of conservative fields, where exact differentials ensured path-independent work in potential theory.^[10]

First-Order Exact Equations

Identification Condition

A first-order differential equation of the form M(x, y) \, dx + N(x, y) \, dy = 0 is exact if and only if \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} throughout the domain where M and N are continuously differentiable.^[1] This condition serves as the necessary and sufficient test for exactness, linking the differential form directly to the existence of a potential function. To prove the necessity of this condition, assume there exists a function F(x, y) such that \frac{\partial F}{\partial x} = [M](/page/M) and \frac{\partial F}{\partial y} = [N](/page/N+), with F twice continuously differentiable. Differentiating the first equation with respect to y yields \frac{\partial^2 F}{\partial y \partial x} = \frac{\partial [M](/page/M)}{\partial y}, while differentiating the second with respect to x gives \frac{\partial^2 F}{\partial x \partial y} = \frac{\partial [N](/page/N+)}{\partial x}. By Clairaut's theorem on the equality of mixed partial derivatives, \frac{\partial [M](/page/M)}{\partial y} = \frac{\partial [N](/page/N+)}{\partial x}.^[11] For sufficiency, suppose \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} holds in a simply connected region R \subseteq \mathbb{R}^2. Integrate M with respect to x, treating y as constant: F(x, y) = \int M(x, y) \, dx + \phi(y), where \phi(y) is an arbitrary function of y. Differentiating this with respect to y produces \frac{\partial F}{\partial y} = \frac{\partial}{\partial y} \left( \int M(x, y) \, dx \right) + \phi'(y). Setting this equal to N gives \phi'(y) = N(x, y) - \frac{\partial}{\partial y} \left( \int M(x, y) \, dx \right). The right-hand side is independent of x due to the exactness condition, allowing integration with respect to y to yield \phi(y). Thus, F(x, y) satisfies both partial derivative requirements, confirming exactness.^[11] This construction relies on the Poincaré lemma, which guarantees the existence of such a potential in simply connected domains under the given continuity assumptions.^[12] The domain must be open and simply connected (i.e., path-connected without holes) for the sufficiency to hold globally; in multiply connected regions, a closed form may not be exact even if the local condition is satisfied.^[11] If \frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x} at any point in the domain, no such F exists, rendering the equation inexact.^[1]

Solution Procedure

To solve a first-order exact differential equation of the form M(x, y) \, dx + N(x, y) \, dy = 0, where the exactness condition \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} holds, the goal is to find a function F(x, y) such that dF = M \, dx + N \, dy, leading to the implicit solution F(x, y) = C, with C a constant.^[1]^[13] The standard procedure begins by integrating M with respect to x, treating y as constant, to obtain

F(x, y) = \int M(x, y) \, dx + g(y),

where g(y) is an arbitrary function of y to be determined.^[1]^[14] Differentiate this expression partially with respect to y:

\frac{\partial F}{\partial y} = \frac{\partial}{\partial y} \left( \int M(x, y) \, dx \right) + g'(y).

Set this equal to N(x, y) and solve the resulting equation for g'(y), which typically yields g'(y) as a function of y alone due to exactness. Integrate g'(y) with respect to y to find g(y), adding any constant of integration (which can be absorbed into C).^[1]^[13] To verify the solution, compute \frac{\partial F}{\partial x} and confirm it equals M(x, y); the exactness condition ensures consistency with \frac{\partial F}{\partial y} = N(x, y).^[14] If the integral of M with respect to x is complex, integrate N with respect to y first instead:

F(x, y) = \int N(x, y) \, dy + h(x),

then differentiate with respect to x, set equal to M, and solve for h(x) by integration. This approach is chosen when one variable yields simpler antiderivatives.^[1]^[15] The resulting F(x, y) = C provides the general implicit solution; if solvable for y explicitly, an explicit form y = f(x, C) may be obtained, though implicit forms are often sufficient and preferred for their direct derivation from the total differential.^[13]^[15]

Higher-Order Exact Equations

Second-Order Case

A second-order ordinary differential equation (ODE) of the form a_2(x, y, y') y'' + a_1(x, y, y') y' + a_0(x, y, y') = 0 is exact if it can be written as the total derivative with respect to x of some function \Psi(x, y, y'), i.e., \frac{d\Psi}{dx} = 0. This means there exists \Psi such that \frac{\partial \Psi}{\partial x} = a_0, \frac{\partial \Psi}{\partial y} = a_1, and \frac{\partial \Psi}{\partial y'} = a_2. The exactness conditions are the compatibility requirements for such a \Psi to exist:

\frac{\partial a_0}{\partial y} = \frac{\partial a_1}{\partial x}, \quad \frac{\partial a_0}{\partial y'} = \frac{\partial a_2}{\partial x}, \quad \frac{\partial a_1}{\partial y'} = \frac{\partial a_2}{\partial y}.

These ensure the mixed partial derivatives of \Psi are equal. To reduce the equation, introduce the substitution v = y', so y'' = \frac{dv}{dx}. The ODE becomes a_2(x, y, v) \frac{dv}{dx} + a_1(x, y, v) v + a_0(x, y, v) = 0, or in differential form,

[a_0(x, y, v) + a_1(x, y, v) v] \, dx + a_2(x, y, v) \, dv = 0.

Denote M(x, y, v) = a_0(x, y, v) + a_1(x, y, v) v and N(x, y, v) = a_2(x, y, v). Treating y as fixed (a parameter), this is a first-order ODE in x and v. It is exact if \frac{\partial M}{\partial v} = \frac{\partial N}{\partial x}, which is one of the overall exactness conditions. If all three conditions hold, the full set ensures consistency upon substituting back dy = v \, dx. The function \Psi is found by successive integration:

\Psi(x, y, v) = \int a_0(\xi, y, v) \, d\xi + \int \left[ a_1(x, \eta, v) - \frac{\partial}{\partial \eta} \int a_0(\xi, \eta, v) \, d\xi \right] d\eta + g(v),

where g(v) is chosen to satisfy \frac{\partial \Psi}{\partial v} = a_2, and similar adjustments for the other integrals if needed. The first integral is then \Psi(x, y, v) = c_1. Substituting v = y' yields the reduced first-order ODE \Psi(x, y, y') = c_1, which is solved using standard first-order techniques; if this equation can be expressed in exact form (e.g., as P(x, y) \, dx + Q(x, y) \, dy = 0 after solving for y'), the exact solution procedure applies directly. For the specific case y'' = f(x, y, y'), rewrite as $1 \cdot y'' + 0 \cdot y' - f(x, y, y') = 0, so a_2 = 1, a_1 = 0, a_0 = -f. The exactness conditions simplify to \frac{\partial (-f)}{\partial y} = \frac{\partial 0}{\partial x} (i.e., \frac{\partial f}{\partial y} = 0) and \frac{\partial (-f)}{\partial y'} = \frac{\partial 1}{\partial x} (i.e., \frac{\partial f}{\partial y'} = 0), with the third trivially satisfied. Thus, the equation is exact if f = f(x). In this case, M = -f(x), N = 1, and the form is -f(x) \, dx + dv = 0, with exactness \frac{\partial (-f)}{\partial v} = 0 = \frac{\partial 1}{\partial x}. Integrating gives \Psi(x, y, v) = v - \int f(x) \, dx = c_1, so y' = \int f(x) \, dx + c_1, a separable (hence exact) first-order equation whose solution is y = \iint f(x) \, dx \, dx + c_1 x + c_2.

General nth-Order Case

In the general case, an nth-order ordinary differential equation (ODE) involving the dependent variable y and its derivatives up to order n is defined to be exact if it can be written in the form dF(x, y, y', \dots, y^{(n-1)}) = 0, where F is a sufficiently smooth function depending on the independent variable x and the first n-1 derivatives of y. This formulation implies that the ODE represents the total differential of an (n-1)-fold integral, enabling successive integrations to yield the general solution directly without additional techniques. The concept originates from early treatments of integrability in differential equations, where exactness ensures the equation is the exterior derivative of a potential function in the space of variables (x, y, y', \dots, y^{(n-1)}).^[16] The condition for exactness generalizes Clairaut's theorem on the equality of mixed partial derivatives to multiple variables. For the differential form associated with the ODE to be exact, the second partial derivatives of F must commute, i.e., \frac{\partial^2 F}{\partial u \partial v} = \frac{\partial^2 F}{\partial v \partial u} for all pairs of variables u, v among

\{[x, y](/page/X&Y), y', \dots, y^{(n-1)}\}

, assuming sufficient smoothness. This commutativity ensures the form is closed, meaning its exterior derivative vanishes, a necessary and sufficient condition for local exactness in the relevant domain. Failure of this condition indicates the need for an integrating factor to render the equation exact.^[17] An exact nth-order ODE admits iterative reduction of order through integration. Integrating the equation once produces an equation of order n-1 in the variables x, y, y', \dots, y^{(n-2)}, which inherits exactness from the original due to the structure of the total differential. This process repeats, successively lowering the order until a solvable first-order exact equation is obtained, from which the full solution follows by standard methods. Each integration introduces an arbitrary constant, yielding the n constants expected in the general solution of an nth-order ODE.^[18] This framework connects to the theory of Pfaffian systems, where the nth-order ODE is reformulated as a system of first-order Pfaffian equations in an extended phase space with variables x, y = p_0, p_1 = y', \dots, p_{n-1} = y^{(n-1)}, including relations dp_k = p_{k+1} \, dx for k = 0, \dots, n-2 and the original ODE for dp_{n-1}. The exactness corresponds to the integrability of this Pfaffian system, governed by the Frobenius theorem, which states that the system is integrable if and only if the ideal generated by the 1-forms is closed under exterior differentiation, i.e., the 2-forms d\omega_i \wedge \omega_j = 0 for all basis 1-forms \omega_i, \omega_j. This provides a rigorous algebraic criterion for exactness in higher dimensions, linking classical ODE theory to modern differential geometry.^[17]^[18]

Applications and Examples

Exact differential equations arise in various contexts where path independence or conservation laws are key. A classic first-order example is the equation (2x + y)\, dx + (x + 2y)\, dy = 0. This is exact since \frac{\partial}{\partial y}(2x + y) = 1 = \frac{\partial}{\partial x}(x + 2y). Integrating the first term with respect to x yields F(x, y) = x^2 + xy + h(y). Differentiating with respect to y gives x + h'(y) = x + 2y, so h'(y) = 2y and h(y) = y^2. The solution is thus x^2 + xy + y^2 = C.^[13] For second-order equations, consider the exact equation y'' + \frac{2}{x} y' = 0 for x > 0. Here, a_2 = 1, a_1 = \frac{2}{x}, a_0 = 0. The conditions hold: \frac{\partial a_0}{\partial y} = 0 = \frac{\partial a_1}{\partial x} = -\frac{2}{x^2} wait, no—actually, this is not general form since a1 depends on x only. But it is the derivative of \frac{1}{x^2} (x^2 y')' = 0, or directly, multiply by x^2: (x^2 y')' =0, so x^2 y' = c_1, y' = c_1 / x^2, y = -c_1 / x + c_2. This illustrates reduction to first-order exact after recognizing the structure.^[1] A third-order linear exact differential equation satisfies the condition where the coefficients p, q, r obey p' = q and q' = r. An example is y''' + 3x^2 y'' + 6x y' + 6y = 0, with p = 3x^2, q = 6x = p', and r = 6 = q'. This exactness allows successive integrations: the first yields a second-order equation y'' + 3x^2 y' + 6x y = c_1, the second reduces to y' + 3x^2 y = c_1 x + c_2, and the third integrates to the solution involving logarithmic terms characteristic of Euler equations, y = c_1 + c_2 \ln x + c_3 (\ln x)^2.^[19] In physics, exact differential equations model conservative vector fields, where a field \mathbf{F} = P \mathbf{i} + Q \mathbf{j} is conservative if \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}, ensuring the line integral \int \mathbf{F} \cdot d\mathbf{r} is path-independent and equals the potential difference f(B) - f(A), with df = P\, dx + Q\, dy. This corresponds to \mathbf{F} = \nabla f, tying directly to exactness. In thermodynamics, exact differentials describe changes in state functions like internal energy U, where dU = đq + đw combines inexact heat đq and work đw into an exact form, path-independent and dependent only on initial and final states, such as dU = T\, dS - P\, dV.^[20]