Function implicitly determined by an equation

Suppose the values ​​of the variables x, y are linked by an equation that will be written symbolically and briefly

H (x; y) = 0 → (1)

If the function y = φ (x) defined in a certain interval <a; b> and by replacing a and in (1), the equation becomes an identity with respect to x, it is called a function implicitly determined by an equation ; in this case po H (x; y). [one]

Summary

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  • 1 Examples
  • 2 Bypass
  • 3 See also
  • 4 Bibliography
  • 5 References

Examples

  • 2x + 3y -5 = 0.
  • 9x 2+ 4y 2 – 36, → (2), which implicitly determines the following two functions:
  1. y = 0.5 (36-9x 25
  2. y = -0.5 (36-9x 25

.. these equations have been obtained by solving equation (2).

However, there are cases when the proposed equation cannot be solved:

  1. x -8y = e x-y
  2. 2+ y 2 = sin (x + y)

The general solutions of many ordinary differential equations are equations that carry functions that are implicitly definable, but not explicit. [2]

Derivation

Example 1

Let be the equation x 3 + y 3 = 3axy, we are going to derive considering y = y (x) and using the derivative rule of the compound function.

3x 2 + 3y 2 y ‘= 3ay + 3axy’ and finally

y ‘= (ay-x 2 ) ÷ (y 2 – ax)

Example 2

Given the equation sin x + cos y = a, find y ‘, we have

cosx + seny · y ‘= 0

y ‘= cosx ÷ sin y

 

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