define exact differential equation
Answers
Step-by-step explanation:
In mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in physics and engineering.
Answer:
In mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in physics and engineering.
Definition
Given a simply connected and open subset D of R2 and two functions I and J which are continuous on D, an implicit first-order ordinary differential equation of the form
{\displaystyle I(x,y)\,\mathrm {d} x+J(x,y)\,\mathrm {d} y=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,[1][2] so that
{\displaystyle {\frac {\partial F}{\partial x}}=I}\frac{\partial F}{\partial x} = I
and
{\displaystyle {\frac {\partial F}{\partial y}}=J.}\frac{\partial F}{\partial y} = J.
The nomenclature of "exact differential equation" refers to the exact differential of a function. For a function {\displaystyle F(x_{0},x_{1},...,x_{n-1},x_{n})}F(x_0, x_1,...,x_{n-1},x_n), the exact or total derivative with respect to {\displaystyle x_{0}}x_{0} is given by
{\displaystyle {\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}}}.}\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}.
Example Edit
The function {\displaystyle F:\mathbb {R} ^{2}\to \mathbb {R} }F:\mathbb{R}^{2}\to\mathbb{R} given by
{\displaystyle F(x,y)={\frac {1}{2}}(x^{2}+y^{2})+c}{\displaystyle F(x,y)={\frac {1}{2}}(x^{2}+y^{2})+c}
is a potential function for the differential equation
{\displaystyle x\,\mathrm {d} x+y\,\mathrm {d} y=0.\,}x\,\mathrm{d}x + y\,\mathrm{d}y = 0.\,