[LON-CAPA-users] Implicit Answer

[Peter Dencker]

Bayu Irjanto wrote:
 > One of the questions would be like
 >
 > Solve y'=t/(y^3-5).
 >
 > And the correct answer for this differential equation is
 >
 > y^4/4-5y=t^2/2+C.


Hi.

An approach using equations as submissions is here, for mathematical 
reasons, really hard:

y^4/4-5y-t^2/2 = C^3-C + 2

and even

t^2y^4/4+y^4/4-5y^3-t^2y^2/2-5t^2y-5y-t^4/2-t^2/2
   = C^5+y^2C^3+t^2C^3-y^4C^2/4+5y*C^2+t^2C^2/2-y^2*C-t^2*C-C-y^6/4

describe the same family, whereas

t^2y^4/4+y^4/4+5y^3+t^2y^2/2-5t^2y-5y-t^4/2-t^2/2
  = C^5-y^2C^3+t^2C^3-y^4C^2/4+5y*C^2+t^2C^2/2+y^2*C-t^2*C-C+y^6/4

does not.

In your case it may be better to use the following reformulation:

"Submit a function R (t, y) in the variables t and y,
  such that for real C
     R (t, y) = C
  describes the family of the regular solutions of  ..."

Remark that even in this formulation 'R (t, y)'  is only determined up 
to an arbitrary non-zero constant factor.


- Peter

-- 
Dr. Peter Dencker
     wissenschaftl. Mitarbeiter

UNIVERSITÄT ZU LÜBECK
     INSTITUT FÜR MATHEMATIK

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     dencker at math.uni-luebeck.de

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