Need reassurance on "implicit" and "explicit" form

In summary, the conversation discusses solving a separable equation, specifically the given differential equation $y' = x^2/y$. After manipulation and taking the integral, the implicit form is found to be $\frac{y^2}{2} = \frac{x^3}{3} + C$, with the explicit form being $y = \pm\sqrt{\frac{2}{3}x^3+C}$. The conversation ends with a confirmation of the explicit form.
  • #1
shamieh
539
0
When dealing with this separable equation for example, if I'm told to solve the given D.E.

$y' = x^2/y$

so after manipulation and taking the integral I got $\frac{y^2}{2} = \frac{x^3}{3} + C$ This is the implicit form correct?

Would the explicit form be $y = \sqrt{\frac{2}{3} x^3 + C}$
 
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  • #2
shamieh said:
When dealing with this separable equation for example, if I'm told to solve the given D.E.

$y' = x^2/y$

so after manipulation and taking the integral I got $\frac{y^2}{2} = \frac{x^3}{3} + C$ This is the implicit form correct?

Correct. :D

shamieh said:
Would the explicit form be $y = \sqrt{\frac{2}{3} x^3 + C}$

I would write:

$y = \pm\sqrt{\frac{2}{3}x^3+C}$
 

What is the difference between "implicit" and "explicit" form?

The main difference between implicit and explicit form is the way information is expressed. In implicit form, information is implied or suggested, while in explicit form, information is stated directly.

How do you convert a function from implicit to explicit form?

To convert a function from implicit to explicit form, you need to solve for the dependent variable. This involves isolating the dependent variable on one side of the equation and rewriting the other side in terms of the dependent variable.

Can a function be written in both implicit and explicit form?

Yes, a function can be written in both implicit and explicit form. In fact, a function can be converted from one form to the other depending on what is most convenient for a particular problem.

What is an example of a function in implicit form?

An example of a function in implicit form is the equation of a circle: x2 + y2 = r2. In this case, the relationship between x and y is implied by the equation, rather than stated explicitly.

How is implicit form useful in mathematics and science?

Implicit form is useful in mathematics and science because it allows for a more general representation of a relationship between variables. It can also make certain calculations and proofs easier, as well as providing a way to graph equations that cannot be easily written in explicit form.

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