- Exact differential equation
In
mathematic s, an exact differential equation or total differential equation is a certain kind ofordinary differential equation which is widely used inphysics andengineering .Definition
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-orderordinary differential equation of the form: I(x, y), mathrm{d}x + J(x, y), mathrm{d}y = 0, ,!
is called exact differential equation if there exists a
continuously differentiable function "F", called the potential function, so that:frac{partial F}{partial x}(x, y) = Iand:frac{partial F}{partial y}(x, y) = J.The nomenclature of "exact differential equation" refers to the exact derivative (or total 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}.
Examples
The function :F(x,y) := frac{1}{2}(x^2 + y^2)is a potential function for the differential equation:xx' + yy' = 0.,
Existence of potential functions
In physical applications the functions "I" and "J" are usually not only continuous but even
continuously differentiable . Schwarz's Theorem (also known as Clairaut's Theorem) then provides us with a necessary criterion for the existence of a potential function. For differential equations defined on simply connected sets the criterion is evensufficient and we get the following theorem:Given a differential equation of the form: I(x, y), dx + J(x, y), dy = 0, ,!with "I" and "J" continuously differentiable on a simply connected and open subset "D" of R2 then a potential function "F" exists if and only if :frac{partial I}{partial y}(x, y) = frac{partial J}{partial x}(x, y).
Solutions to exact differential equations
Given an exact differential equation defined on some simply connected and open subset "D" of R2 with potential function "F" then a differentiable function "f" with (x, "f"("x")) in "D" is a solution
if and only if there existsreal number "c" so that:F(x, f(x)) = c.,For an
initial value problem :y(x_0) = y_0,we can locally find a potential function by:F(x,y) = int_{x_0}^x I(t,y_0) dt + int_{y_0}^y J(x,t) dt.Solving :F(x,y) = c,for "y", where "c" is a real number, we can then construct all solutions.
See also
*
exact differential
*total derivative
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