Elimination of Arbitrary Constant

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SUMMARY

The discussion focuses on the mathematical problem of eliminating the arbitrary constant from the equation y = 8a³ / (x² + 4a²). Participants clarify that the equation is already solved for y and guide the user through the process of implicit differentiation. The key steps involve rearranging the equation and applying implicit differentiation to derive relationships between the variables. Ultimately, the goal is to eliminate the arbitrary constant 'a' from the equation.

PREREQUISITES
  • Understanding of implicit differentiation
  • Familiarity with algebraic manipulation of equations
  • Knowledge of how to handle arbitrary constants in equations
  • Basic calculus concepts
NEXT STEPS
  • Study the principles of implicit differentiation in calculus
  • Practice solving equations with arbitrary constants
  • Explore algebraic techniques for rearranging equations
  • Learn about the applications of implicit differentiation in real-world problems
USEFUL FOR

Students studying calculus, mathematics educators, and anyone interested in mastering the techniques of implicit differentiation and solving equations with arbitrary constants.

kayella19
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y= 8a³ / x²+4a²
 
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Re: Any one who can solve this?

Hello and welcome to MHB, kayella19! (Wave)

What you've posted isn't a differential equation...and the equation appears to already be solved for y...what is the entire question as it was given to you?
 
Re: Any one who can solve this?

Eliminate the arbitrary constant.
 
Re: Any one who can solve this?

Anyone pleasee? I need your help with this eliminating arbitrary constant. y= 8a³/ (x²+4a²)
 
Re: Any one who can solve this?

kayella19 said:
Anyone pleasee? I need your help with this eliminating arbitrary constant. y= 8a³/ (x²+4a²)

We are given:

$$y=\frac{8a^3}{x^2+4a^2}$$

This implies:

$$\left(x^2+4a^2\right)y=8a^3$$

Implicit differentiation yields:

$$x^2+4a^2=-\frac{2xy}{y'}$$

Hence:

$$-\frac{2xy}{y'}=\frac{8a^3}{y}$$

This implies:

$$\frac{xy^2}{y'}=-4a^3$$

What does implicit differentiation yield?
 
How is that happen? What does implicit differentiation implies?

- - - Updated - - -

The arbitrary constant is'nt eliminated too.
 
kayella19 said:
...The arbitrary constant is'nt eliminated too.

Yes, I left a little bit of work for you to do to complete the problem. :D
 

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