Exact Equations

The Hidden Structure

Some differential equations are more structured than they first appear. Consider an equation in the form:

At first glance, this looks like any other first-order equation. But sometimes, the left side is secretly the total differential of some function $F(x, y)$. When that happens, the equation is simply saying $dF = 0$, which means $F$ is constant along solutions.

This is the essence of an exact equation: a differential equation that is really just $dF = 0$ in disguise.

Total Differentials from Multivariable Calculus

Recall from multivariable calculus that if $F(x, y)$ is a function of two variables, its total differential is:

The total differential captures how $F$ changes when both $x$ and $y$ change by small amounts. If a particle moves along a curve where $x$ changes by $dx$ and $y$ changes by $dy$, then $F$ changes by approximately $dF$.

Now here is the key insight. If we have an equation $M\,dx + N\,dy = 0$ and there exists a function $F$ such that:

then the equation becomes $dF = 0$. This means $F$ is constant along any solution curve.

The surface shows $F(x, y)$. The level curves are where $F$ is constant. When we set $dF = 0$, we are asking: which curves in the $xy$-plane keep $F$ unchanged? The answer is precisely the level curves of $F$.

The Exactness Test

How do we know if an equation $M\,dx + N\,dy = 0$ is exact? We need to check whether there exists a function $F$ with $F_x = M$ and $F_y = N$.

Here is where a beautiful fact from calculus comes to the rescue. If $F$ has continuous second partial derivatives, then the mixed partials are equal:

This means $\frac{\partial}{\partial y}(F_x) = \frac{\partial}{\partial x}(F_y)$, or equivalently:

This is the exactness test. If the equation is exact, these partial derivatives must be equal. Conversely, in a simply connected region, if these partials are equal, the equation is exact.

Enter any functions $M$ and $N$. The checker computes the partial derivatives and tells you whether the equation is exact. When $\partial M/\partial y = \partial N/\partial x$, a potential function $F$ exists.

Why Does This Test Work?

The logic is elegant. If $F$ exists with $F_x = M$ and $F_y = N$, then:

The equality of mixed partials guarantees that the exactness test passes. Going the other direction (proving existence when the test passes) requires a theorem about conservative vector fields, but the practical upshot is simple: check if the cross-partials match.

Finding the Potential Function

Once we know an equation is exact, we need to find the function $F(x, y)$ such that $F_x = M$ and $F_y = N$. The method has two steps.

Step 1: Integrate with respect to one variable.

Start with $F_x = M$ and integrate with respect to $x$:

The "constant" of integration is actually a function $g(y)$ because when we differentiate $F$ with respect to $x$, any function of $y$ alone would vanish.

Step 2: Determine the unknown function.

Now use the condition $F_y = N$. Differentiate our expression for $F$ with respect to $y$:

This lets us solve for $g'(y)$, and then integrate to find $g(y)$.

Follow along as the method unfolds step by step. Enter an exact equation and watch the integration process reveal the potential function $F$.

Solutions Are Level Curves

Once we have $F(x, y)$, the general solution to $M\,dx + N\,dy = 0$ is:

where $C$ is an arbitrary constant. Each choice of $C$ gives a different solution curve, and together they form a family of level curves of $F$.

This is geometrically beautiful. The solution curves are the contours of the potential function. They never cross (except at singular points), and they tile the plane into nested families.

Explore how different values of $C$ trace out different solution curves. The gradient $\nabla F$ is always perpendicular to the level curves, which connects to the fact that the direction field is tangent to solutions.

A Worked Example

Consider the equation:

Check exactness: Here $M = 2xy + 3$ and $N = x^2 + 4y$.

They are equal, so the equation is exact.

Find F: Integrate $M$ with respect to $x$:

Differentiate with respect to $y$:

Set this equal to $N = x^2 + 4y$:

So the potential function is $F(x, y) = x^2 y + 3x + 2y^2$, and the general solution is:

When Equations Are Not Exact: Integrating Factors

What if $\partial M/\partial y \neq \partial N/\partial x$? All is not lost. Sometimes we can multiply the entire equation by a function $\mu(x, y)$ called an integrating factor to make it exact.

The modified equation $\mu M\,dx + \mu N\,dy = 0$ is exact if:

Finding $\mu$ in general is as hard as solving the original equation. But there are special cases:

If $\mu$ depends only on $x$: The condition becomes

If the right side depends only on $x$, we can find $\mu(x)$ by integration.

If $\mu$ depends only on $y$: The condition becomes

If the right side depends only on $y$, we can find $\mu(y)$ by integration.

This technique extends the reach of the exact equation method significantly.

Connection to Physics

Exact equations arise naturally in physics and engineering. When $M\,dx + N\,dy = 0$ is exact, the vector field $(M, N)$ is conservative. The function $F$ is the potential energy, and solutions follow paths of constant energy.

In thermodynamics, exact differentials appear as state functions. The internal energy $U$ of a system satisfies $dU = \delta Q - \delta W$, but only certain combinations form exact differentials. Recognizing exactness tells us which quantities are true state functions, independent of the path taken.

In fluid mechanics, stream functions and velocity potentials satisfy exact equations. The level curves of a stream function are streamlines along which fluid particles travel.

Key Takeaways

• An exact equation $M\,dx + N\,dy = 0$ is secretly the total differential $dF = 0$ of some potential function $F(x, y)$
• The exactness test: check if $\partial M/\partial y = \partial N/\partial x$ (equality of mixed partials)
• To find $F$: integrate $M$ with respect to $x$, then use $F_y = N$ to determine the unknown function of $y$
• Solutions are level curves $F(x, y) = C$, forming a family of contours of the potential function
• Non-exact equations can sometimes be made exact by multiplying by an integrating factor $\mu$
• Exact equations connect to conservative vector fields, potential energy, and state functions in physics