Peter Dencker dencker at math.uni-luebeck.de
Thu Mar 12 20:14:25 EDT 2015

```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

Ratzeburger Allee 160
23562 Lübeck

Tel +49 451 500 4254
Fax +49 451 500 3373
dencker at math.uni-luebeck.de

www.math.uni-luebeck.de
```