Figuring out equivalent constants

Hello everyone,

I have two equations which are equal to the other, using undefined constants, and basically I would like help figuring out what the constants on the right-hand side are equal to in terms of the constants on the left hand side.

the equation is a little long and has lots of constants but is not too complicated:

a*b/(1+(b*x-c)^2) - d*e/(1+(e*x+f)^2) + g*i/(h-i*x) + j(2m*x-l)/(k-l*x+m*x^2) + n(p+2q*x)/(o+p*x+q*x^2) = r / (s/(t*x+u) - v*x^4)

[what are r, s, t, u and v equal to in terms of a-q]

math software?

thank you in advance if you can help, I'd appreciate it.

Re: Figuring out equivalent constants

I assume that the equation should be true for all real x?

$\displaystyle \frac{ab}{1+(bx-c)^2} - \frac{de}{1+(ex+f)^2} + \frac{gi}{h-ix} + \frac{j(2mx-l)}{k-lx+mx^2} + \frac{n(p+2qx)}{o+px+qx^2} = \frac{r}{\frac{s}{tx+u} - vx^4}$

Two parameters at the right side are redundant. Apart from special cases, you can always set r=s=1, for example. In general, I would not expect that there are r,s,t,u,v to get both sides equal for all x.

Where does the equation come from?

Re: Figuring out equivalent constants

Hello mfb,

This equation comes from the integration of the right expression, and since i couldn't do it manually I got the answer using Mathematica (not very good with it), but i wanted to know where the coefficients were coming from, so I thought the best way to proceed was to differentiate the integration so that the two sides would be equal.

using numbers this is the expression:

∫0.0024185/(2500/(0.485x+266)-8.3168 ×10^(-11) x^4 ) dx

=

-0.0821125 arctan(0.171695-0.00193589 x) - 0.0280283 arctan(0.00344281 x+2.10174)-0.0540371 ln(4.1169-0.00833949 x) +0.0181571 ln(0.0000695471 x^2-0.0123363x +19.1044)+0.00886143 ln( 0.0000695471x^2 +0.084913x+31.7859) + c

I'm doing this for a physics project, btw, so for a general equation i'd like to know where the coefficients end up after the integration.

sorry for the big muddle

[now i realize that in the first fraction it should actually be (c-bx)^2]

Re: Figuring out equivalent constants

I would expect that Mathematica can give you the coefficients.

However, why do you want to express r,s,t,u,v in terms of the other parameters, if you already know them and need a-q?

Quote:

[now i realize that in the first fraction it should actually be (c-bx)^2]

As you square it, the sign does not matter.

Using t=v=1 at the right side (you can always get this form), this can be written as $\displaystyle \frac{r(x+u)}{s-ux^4-x^5}$

For a partial fraction decomposition, you would have to know the solutions of 0=s-ux^4-x^5, which should not have a closed form for arbitrary parameters s and u. In other words, I doubt that this integral can be solved in a meaningful way for all parameters.

Re: Figuring out equivalent constants

r,s,t,u,v are variables that remain constant during the process i'm trying to describe, so i have values for them, but i'd like to be able to insert them in the arctan and log function and thus get a general equation.

But if you say it can't be done then that's too bad.

1 Attachment(s)

Re: Figuring out equivalent constants

Hold on, I somehow managed to get an answer using the wolfram software. It is really complicated but it's something:

Attachment 24164

any opinion?

Re: Figuring out equivalent constants

This looks like the case u=0, which is much simpler.

Instead of b/(cd), you can simplify the equation to set this to 1. This reduces the size of the solution.