Exact Equations
Generally speaking, exact equations are equations that can be broken down into the two partial derivatives of some function {$ \Psi(x,y) $}. In the example below, note also that this function is neither linear nore separable.
{$ 2x + y^2 + 2xyy’ = 0 $}
{$ \Psi(x,y) = x^2 + xy^2 $}
{$ 2x + y^2 = \frac{\partial \Psi}{\partial x}$}, {$ 2xy = \frac{\partial \Psi}{\partial y} $}
{$ \frac{\partial \Psi}{\partial x} + \frac{\partial \Psi}{\partial y} \frac{dy}{dx} = 0$}
From the above, we can extrapolate, using the chain rule and assuming that y is a function of x-
{$ \frac{d\Psi}{dx} = \frac{d}{dx}(x^2 + xy^2) = 0$}
Then integrate..
{$ \Psi(x,y) = x^2 + xy^2 = c$}
If this general pattern follows for an equation, it is said to be exact.
It is not always this easy to find {$\Psi$}, though, nor is it always easy to see if it exists. There is a theorem stating that in a simply connected region, the equation {$M(x,y) + N(x,y)y’ = 0$} is exact if the partial derivatives are equal- {$M_y(x,y) = N_x(x,y)$}. One can connect this to the previous problem solution by realizing that this is because {$M(x,y) $} and {$N(x,y)$} must be equal to the partial of {$\Psi$} with respect to x and y, respectively.
After this point, it is a matter of backsolving.
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