Determine whether or not each of the equations is exact. If it is exact, find the solution. (3 x

step 1 Here we are given the first quarter differential equation 3x squared minus 2 xY plus 2 dx plus 6

Determine whether or not each of the equations is exact. If...

Key Concepts

Potential Function Method

When a differential equation is found to be exact, one can find a potential function ?(x, y) such that ?? = M and ?? = N. This is typically done by integrating M with respect to x and then determining the remaining function of y by comparing with N, or vice versa. The general solution to the differential equation is then given by ?(x, y) = constant.

Exact Differential Equation

An exact differential equation is one that can be expressed as the total differential of a function of two variables. In other words, there exists a function ?(x, y) such that the differential equation can be written as d? = ?? dx + ?? dy = 0. This property allows for the integration of the equation by finding ?(x, y) and setting it equal to a constant.

Test for Exactness

To determine if a differential equation of the form M(x, y) dx + N(x, y) dy = 0 is exact, you compare the partial derivative of M with respect to y (denoted M?) with the partial derivative of N with respect to x (denoted N?). The equation is exact if M? = N? for all (x, y) in the domain of interest.

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