Category:Examples of Exact Differential Equation
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This category contains examples of exact differential equations.
Definition 1
An exact differential equation is a first order ordinary differential equation in which the total differential is equal to zero:
- $\dfrac {\partial f} {\partial x} \rd x + \dfrac {\partial f} {\partial y} \rd y = 0$
Definition 2
Let a first order ordinary differential equation be expressible in this form:
- $\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$
such that $M$ and $N$ are not homogeneous functions of the same degree.
However, suppose there happens to exist 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.
Then the expression $M \rd x + N \rd y$ is called an exact differential, and the differential equation is called an exact differential equation.
Pages in category "Examples of Exact Differential Equation"
The following 19 pages are in this category, out of 19 total.
F
- First Order ODE/(1 over x^3 y^2 + 1 over x) dx + (1 over x^2 y^3 - 1 over y) dy = 0
- First Order ODE/(2 x y^3 + y cosine x) dx + (3 x^2 y^2 + sine x) dy
- First Order ODE/(3 x^2 over y^4 - 1 over y^2) dy - 2 x over y^3 dx = 0
- First Order ODE/(exp x - 3 x^2 y^2) y' + y exp x = 2 x y^3
- First Order ODE/(sine x sine y - x e^y) dy = (e^y + cosine x cosine y) dx
- First Order ODE/(x + (2 over y)) dy + y dx = 0
- First Order ODE/(x exp y + y - x^2) dy = (2 x y - exp y - x) dx
- First Order ODE/(y + y cosine x y) dx + (x + x cosine x y) dy = 0
- First Order ODE/(y - 1 over x) dx + (x - y) dy = 0
- First Order ODE/(y - x^3) dx + (x + y^3) dy = 0
- First Order ODE/(y over x^2) dx + (y - 1 over x) dy = 0
- First Order ODE/(y^2 exp x y + cosine x) dx + (exp x y + x y exp x y) dy = 0
- First Order ODE/-1 over y sine x over y dx + x over y^2 sine x over y dy
- First Order ODE/1 over x^3 y^2 dx + (1 over x^2 y^3 + 3 y) dy = 0
- First Order ODE/Cosine (x + y) dx = sine (x + y) dx + x sine (x + y) dy
- First Order ODE/dx = (y over (1 - x^2 y^2)) dx + (x over (1 - x^2 y^2)) dy
- First Order ODE/exp x sine y dx + exp x cos y dy = y sine x y dx + x sine x y dy
- First Order ODE/exp y dx + (x exp y + 2 y) dy = 0
- First Order ODE/y' ln (x - y) = 1 + ln (x - y)