Necessary and Sufficient Condition for Exactness of a Differential Equation

To determine whether a differential equation of the form Mdx + Ndy = 0 is exact, we need to check for a necessary and sufficient condition. The condition involves the partial derivatives of M and N with respect to x and y.

Necessary Condition: For a differential equation Mdx + Ndy = 0 to be exact, it is necessary that the partial derivative of M with respect to y (dM/dy) is equal to the partial derivative of N with respect to x (dN/dx).

Mathematically, the necessary condition can be expressed as follows:

dM/dy = dN/dx

If this condition is not satisfied, then the differential equation is not exact.

Sufficient Condition: The sufficient condition for a differential equation Mdx + Ndy = 0 to be exact involves the concept of an integrating factor.

If the necessary condition is satisfied, i.e., dM/dy = dN/dx, then a function μ(x, y) can be found such that when the differential equation is multiplied by this integrating factor, it becomes an exact differential equation.

The integrating factor, denoted by μ(x, y), is given by the following formula:

μ(x, y) = exp(∫(dN/dx - dM/dy) dx)

where ∫(dN/dx - dM/dy) dx represents the indefinite integral of (dN/dx - dM/dy) with respect to x.

Once the integrating factor is determined, the exact differential equation is obtained by multiplying both sides of the original equation by μ(x, y).

M(x, y)dx + N(x, y)dy = 0 (original equation)

μ(x, y)M(x, y)dx + μ(x, y)N(x, y)dy = 0 (exact differential equation)

By finding the integrating factor and multiplying the original equation by it, we can convert a non-exact differential equation into an exact one.

Proof:

The proof of the necessary and sufficient condition for exactness of a differential equation involves the application of partial derivatives and the concept of total differentials. However, providing a detailed proof within the given word limit is not feasible.

To understand the proof in detail, it is recommended to refer to textbooks or online resources specifically dedicated to the topic of exact differential equations. These resources will provide step-by-step derivations and explanations of the necessary and sufficient condition.