Exam Questions - Exact equations (integrating factors)

Exact Differential Equations

Exact differential equations are a specific category of first-order ordinary differential equations where an integrating factor can be found that makes the equation exact.
To classify an equation as exact, it should correspond to the form Mdx + Ndy = 0, where M and N are expressions of x and y.
If ∂M/∂y equals ∂N/∂x, the equation is said to be exact. Meaning, M and N are the partial derivatives of a function F(x, y).

One crucial feature of an exact differential equation is the integrating factor. This is a function that multiplies an inexact equation to make an exact one.
Identifying an integrating factor can be complex, and often requires advanced methods or guesswork.

Solve the exact equation by evaluating the two integrals: ∫M dx and ∫N dy.
If an equation is exact, both integrals should lead to the same function, since M and N are the derivatives of the same function.
Compare the two integrals and find the common or overlapping part. This will give you the sought function F(x, y).

You will often find the concept of conservation of energy relevant when working with exact differential equations. Consider situations involving mechanical energy, which includes kinetic energy, potential energy, and possible external forces.

The integrating factor plays a pivotal role in transforming an inexact differential into an exact process.
Always identify whether the differential equation is exact by checking if the given equation satisfies the condition ∂M/∂y = ∂N/∂x.
Contrast the integrals derived from M and N. They must lead to the same function if the initial equation is exact.
Always keep an eye out for potential applications in physics or engineering, especially those involving the conservation of energy.