Final answer:
An exact equation can be verified by checking if the cross partial derivatives of M and N are equal, where Mdx + Ndy = 0 represents an exact differential. To find the corresponding function f(x,y), one integrates M and N with respect to x and y, respectively, keeping in mind the constants of integration.
Step-by-step explanation:
To tell if an equation is exact, you need to verify if the equation can be expressed in the form M(x, y)dx + N(x, y)dy = 0, where M and N are the partial derivatives of some function f(x, y) with respect to x and y, respectively. Checking exactness typically involves ensuring that ∂M/∂y = ∂N/∂x. If the equation is exact, then there exists a function f(x, y) such that df = Mdx + Ndy. To find the function f(x, y), you would integrate M with respect to x and N with respect to y, while considering the constants of integration as functions of the other variable.
Finding the Function f(x,y)
The process usually begins with integrating M(x, y) with respect to x, keeping in mind that the constant of integration could be a function of y. Then integrate N(x, y) with respect to y, watching for constants of integration that could depend on x. Comparing the two results will typically provide the necessary insights to determine the full function f(x, y). You may combine terms and solve for any functions that were previously considered as constants of integration.