Exact Differential Equation/Examples

Examples of Exact Differential Equation

This page gathers examples of exact differential equations of the first order:

$\map M {x, y} \rd x + \map N {x, y} \rd y = 0$

where there exists a function $\map f {x, y}$ such that:

$\dfrac {\partial f} {\partial x} = M, \dfrac {\partial f} {\partial y} = N$

such that the second partial derivatives of $f$ exist and are continuous.

$e^y \rd x + \paren {x e^y + 2 y} \rd y = 0$

is an exact differential equation with solution:

$x e^y + y^2 = C$

$\paren {x + \dfrac 2 y} \rd y + y \rd x = 0$

is an exact differential equation with solution:

$x y + 2 \ln y = C$

$\paren {y - x^3} \rd x + \paren {x + y^3} \rd y = 0$

is an exact differential equation with solution:

$4 x y - x^4 + y^4 = C$

$\paren {y + y \cos x y} \rd x + \paren {x + x \cos x y} \rd y = 0$

is an exact differential equation with solution:

$x y + \sin x y = C$

$\paren {\sin x \sin y - x e^y} \rd y = \paren {e^y + \cos x \cos y} \rd x$

is an exact differential equation with solution:

$\sin x \cos y + x e^y = C$

$-\dfrac 1 y \sin \dfrac x y \rd x + \dfrac x {y^2} \sin \dfrac x y \rd y = 0$

is an exact differential equation with solution:

$\dfrac x y = C$

$\paren {2 x y^3 + y \cos x} \rd x + \paren {3 x^2 y^2 + \sin x} \rd y = 0$

is an exact differential equation with solution:

$x^2 y^3 + y \sin x = C$

$\d x = \dfrac y {1 - x^2 y^2} \rd x + \dfrac x {1 - x^2 y^2} \rd y$

is an exact differential equation with solution:

$\map \ln {\dfrac {1 + x y} {1 - x y} } - 2 x = C$