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Free ebook tinyurl.com/EngMathYT How to solve exact differential equations. Given a simply connected and open subset D of R2 and two functions I and J which are continuous on D then an implicit first-order ordinary differential equation of the form

I(x, y)\, \mathrm{d}x + J(x, y)\, \mathrm{d}y = 0, \,\!

is called an exact differential equation if there exists a continuously differentiable function F, called the potential function, so that

\frac{\partial F}{\partial x}(x, y) = I

and

\frac{\partial F}{\partial y}(x, y) = J.

The nomenclature of "exact differential equation" refers to the exact derivative of a function. For a function F(x_0, x_1,...,x_{n-1},x_n), the exact or total derivative with respect to x_0 is given by

\frac{\mathrm{d}F}{\mathrm{d}x_0}=\frac{\partial F}{\partial x_0}+\sum_{i=1}^{n}\frac{\partial F}{\partial x_i}\frac{\mathrm{d}x_i}{\mathrm{d}x_0}.

I(x, y)\, \mathrm{d}x + J(x, y)\, \mathrm{d}y = 0, \,\!

is called an exact differential equation if there exists a continuously differentiable function F, called the potential function, so that

\frac{\partial F}{\partial x}(x, y) = I

and

\frac{\partial F}{\partial y}(x, y) = J.

The nomenclature of "exact differential equation" refers to the exact derivative of a function. For a function F(x_0, x_1,...,x_{n-1},x_n), the exact or total derivative with respect to x_0 is given by

\frac{\mathrm{d}F}{\mathrm{d}x_0}=\frac{\partial F}{\partial x_0}+\sum_{i=1}^{n}\frac{\partial F}{\partial x_i}\frac{\mathrm{d}x_i}{\mathrm{d}x_0}.