Exact Equations

You will learn how to recognize exact differential equations, verify they are exact using partial derivatives, and solve them by finding the potential function.

Some first-order differential equations can be written in differential form:

$$M(x, y) \, dx + N(x, y) \, dy = 0$$

This equation is called exact if there exists a function $\phi(x, y)$ such that:

$$d\phi = M \, dx + N \, dy$$

A quick test tells us whether such a $\phi$ exists:

$$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$

If this condition holds, we can reconstruct $\phi(x, y)$ by integrating $M$ with respect to $x$ (treating $y$ as constant), then adding the remaining terms from $N$.

The solution is then simply:

$$\phi(x, y) = C$$

The visual above shows the level curves of the potential function $\phi(x, y) = x^2 + xy + \frac{3}{2}y^2$. Each ellipse is a solution curve where $\phi$ is constant.

Example 1: Checking Exactness

Determine whether the equation $(2x + y) \, dx + (x + 3y) \, dy = 0$ is exact.

Solution: Here $M = 2x + y$ and $N = x + 3y$.

$$\frac{\partial M}{\partial y} = 1, \quad \frac{\partial N}{\partial x} = 1$$

They are equal, so the equation is exact.

Example 2: Solving an Exact Equation

Solve $(2x + y) \, dx + (x + 3y) \, dy = 0$.

Solution: Integrate $M$ with respect to $x$:

$$\phi(x, y) = x^2 + xy + h(y)$$

Now take the partial with respect to $y$ and match to $N$:

$$\frac{\partial \phi}{\partial y} = x + h'(y) = x + 3y$$

So $h'(y) = 3y$ which gives $h(y) = \frac{3}{2}y^2$.

Final potential function: $\phi(x, y) = x^2 + xy + \frac{3}{2}y^2 = C$

Example 3: With Initial Condition

Solve the exact equation $(y + e^x) \, dx + (x + \cos y) \, dy = 0$ with $y(0) = 0$.

Solution: It is exact. Integrating gives:

$$\phi(x, y) = xy + e^x + \sin y = C$$

Using $y(0) = 0$: $0 + 1 + 0 = C \implies C = 1$

Implicit solution: $xy + e^x + \sin y = 1$

Exact equations appear naturally in physics when working with conservative force fields. For example, finding the potential energy function from a force is essentially solving an exact equation. They also show up in thermodynamics (relating internal energy, entropy, and temperature) and in fluid dynamics when modeling irrotational flow. In engineering, they help solve problems involving exact differentials in electrical networks and heat transfer.