separation of variables involving a quadratic

canadianicon25 · **Posted:** Tue, 7 Aug 2012 21:53:01 UTC

given problem:

dx/dt=k(3-x)(2-x)

x(0)=1

I have a worked solution for this problem and it uses partial fractions as a key method.

My question is whether this problem can be solved without partial fractions, ie: combining terms to produce a quadratic in x.

A yes or no is what I am looking for. If yes I will attempt to work the problem.

outermeasure · **Posted:** Wed, 8 Aug 2012 03:09:44 UTC

canadianicon25 wrote:

given problem:

dx/dt=k(3-x)(2-x)

x(0)=1

I have a worked solution for this problem and it uses partial fractions as a key method.

My question is whether this problem can be solved without partial fractions, ie: combining terms to produce a quadratic in x.

A yes or no is what I am looking for. If yes I will attempt to work the problem.

Recall $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\tanh^{-1}(x)=\frac{1}{1-x^2}$ .

Soroban · **Posted:** Wed, 12 Sep 2012 01:36:18 UTC

Hello, canadianicon25!

Quote:

$\dfrac{dx}{dt} \:=\:k(3-x)(2-x)\quad x(0)\,=\,1$

I have a worked solution for this problem and it uses partial fractions as a key method.

My question is whether this problem can be solved without partial fractions,
ie: combining terms to produce a quadratic in x.

A yes or no is what I am looking for.
If yes, I will attempt to work the problem.

We can also complete-the-square.

$(x-3)(x-2) \;=\;x^2 - 5x + 6$

. . . . . . . . . . . . $=\:\left(x^2 - 5x + \frac{25}{4}\right) + \left(6 - \frac{25}{4}\right)$

. . . . . . . . . . . . $=\;\left(x - \frac{5}{2}\right)^2 - \left(\frac{1}{2}\right)^2$

Then the left side has the form: . $\displaystyle\int\frac{du}{u^2-a^2} \;=\;\frac{1}{2a}\ln\left|\frac{u-a}{u+a}\right| + C$

Matt · **Joined:** Wed, 1 Oct 2003 04:45:43 UTC **Posts:** 9642

Doesn't the derivation of that integration rule use partial fractions?

Shadow · **Posted:** Wed, 12 Sep 2012 06:38:29 UTC

Matt wrote:

Doesn't the derivation of that integration rule use partial fractions?

I think so, but it's probably in some table as well, which is probably what Soroban surpressed.

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separation of variables involving a quadratic

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