Exact differential equations

Exact differential equations are solved by finding a scalar potential that generates the multiplying functions of the first order ODE. Hence a first-order ODE is exact if and only if such a potential exists, i.e. the field $⃗ 𝐅$ (defined below) is conservative.

Consider a function of the form

𝑀 (𝑥, 𝑦) + 𝑁 (𝑥, 𝑦) 𝑑 𝑦 𝑑 𝑥 = 0

We rewrite this in terms of a covector field $⃗ 𝐅$ and a function $⃗ 𝐯 : 𝑥 \mapsto [𝑥 𝑦 (𝑥)] 𝖳$ ¹ so that

(⃗ 𝐅 \circ ⃗ 𝐯) (𝑡) 𝐷 𝑣 (𝑡) = 0

If $⃗ 𝐅$ is conservative, via the multivariable chain rule this can be rewritten as

0 = (𝐷 Ψ \circ ⃗ 𝐯) (𝑥) 𝐷 ⃗ 𝐯 (𝑥) = 𝐷 (Ψ \circ ⃗ 𝐯) (𝑥)

Since this is a function of a single variable, both sides of the equation can be integrated giving

0 = \int 𝑑 𝑑 𝑥 (Ψ \circ ⃗ 𝐯) (𝑥) 𝑑 𝑥 = (Ψ \circ ⃗ 𝐯) (𝑥) + 𝐶 = 𝜓 ([𝑥 𝑦 (𝑥)]) + 𝐶

Which can then be rearranged for $𝑦 (𝑥)$ .²

It follows from this exploration that the solutions to an exact equation are just the Level set of its generating potential, or equivalently that the solutions correspond to each path through the field $⃗ 𝐅$ for which there is no work done in any segment.

Integrating factor

Sometimes a first order ODE will be almost exact, in that it can be made exact via an integrating factor $𝜇 (𝑥, 𝑦)$ , in a similar vein to Integrating factor. If such a $𝜇$ exists, then $(𝜇 ⃗ 𝐅) (⃗ 𝐯)$ is conservative, and hence we get the following first order partial differential equation

𝜕 𝜇 𝜕 𝑥 𝑁 - 𝜕 𝜇 𝜕 𝑦 𝑀 + 𝜇 (𝜕 𝑁 𝜕 𝑥 - 𝜕 𝑀 𝜕 𝑦) = 0

Which is unfortunately usually just as difficult to solve if not more so than the original differential equation.³ However, if we assume that $𝜇$ is only a function of $𝑥$ or only a function of $𝑦$ , and test for each of these cases.

Higher order

[?] Exact differential equations can be extended to arbitrary higher orders,⁴ which is somehow related to adjointness?

Practice problems

2017. Elementary differential equations and boundary value problems, p. 75 (§2.6 problems 1–10)

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We assume $𝑦$ to be a function of $𝑥$ . ↩
The above explanation of exact equations differs slightly from the conventional one: it has been shaped by my preference for explicit functions and the matrix-based chain rule. ↩
2017. Elementary differential equations and boundary value problems, p. 74 ↩
2017. Elementary differential equations and boundary value problems, pp. 119–120 (§3.2 problems 31ff.) ↩

Exact differential equations

Integrating factor

Higher order

Practice problems

Footnotes