List of basic formulas and concepts related to differential equations in total differentials:

  1. 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 ).
  2. 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}
    ]
  3. 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.
  4. 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.
  5. 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.

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