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

[ hide ]

• 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 ²) ⁵
2. y = -0.5 (36-9x ²) ⁵

.. 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. x 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