List of basic formulas and concepts related to differential equations in total differentials:
- The general equation of the total differential is:
[
M(x, y)dx + N(x, y)dy = 0
]
where ( M ) and ( N ) are functions that depend on ( x ) and ( y ). - Total differential condition:
The equation ( M(x, y)dx + N(x, y)dy = 0 ) is a total differential if the condition is satisfied:
[
\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}
] - Integration of the total differential:
If the equation is total, then its solution can be found by integrating ( M ) and ( N ):
[
\frac{\partial F}{\partial x} = M(x, y), \quad \frac{\partial F}{\partial y} = N(x, y)
]
where ( F(x, y) = C ) is the general solution of the equation. - Example of a total differential:
If ( M(x, y) = 2xy ) and ( N(x, y) = x^2 + y^2 ), then:
[
2xy \, dx + (x^2 + y^2) \, dy = 0
]
We check the condition:
[
\frac{\partial M}{\partial y} = 2x, \quad \frac{\partial N}{\partial x} = 2x
]
The condition is met, which means the equation is total. - Properties of the total differential:
- If ( F(x, y) = C ), then ( dF = 0 ).
- The total differential retains the properties of independence from the integration path.