One last Diff Eq

**vodka** · July 12th 2008, 11:07 AM

$y' = -2(3y +4)$

It is separable im lost in how to get the answer,

$y = -4/3 + Ce^-6x$

I tried and made it to $dy = (-6y - 8)dx$ but im unsure as how to get my dy/y for 6lny sigh now im really getting confused. Anyone feel free to shed some light on the case in point.

**galactus** · July 12th 2008, 11:22 AM

Write as

$\frac{dy}{dx}+6y=-8$

The integrating factor is $e^{\int 6dx}=e^{6x}$

$\frac{d}{dx}[e^{6x}y]=-8e^{6x}$

Integrate:

$\int\frac{d}{dx}[e^{6x}y]=-8\int e^{6x}dx$

$e^{6x}y=\frac{-4}{3}e^{6x}+C$

$\boxed{y=\frac{-4}{3}+Ce^{-6x}}$

**cyph1e** · July 12th 2008, 11:25 AM

$\frac{dy}{-6y-8}=dx$

Integrating both sides yields

$-\frac 1 6 \ln(-6y-8)=x+C_1$

Exponentiating:

$-6y-8=C_2e^{-6x}$ where $C_2=e^{-6C_1}$

And finally:

$y=Ce^{-6x}-\frac 4 3$ where $C=-\frac{C_2}{6}$

**vodka** · July 12th 2008, 11:30 AM

Originally Posted by cyph1e

$\frac{dy}{-6y-8}=dx$

Integrating both sides yields

$-\frac 1 6 \ln(-6y-8)=x+C_1$

Exponentiating:

$-6y-8=C_2e^{-6x}$ where $C_2=e^{-6C_1}$

And finally:

$y=Ce^{-6x}-\frac 4 3$ where $C=-\frac{C_2}{6}$

Ahh that makes an little more sense Good Work!

**topsquark** · July 12th 2008, 11:32 AM

Originally Posted by vodka

$y' = -2(3y +4)$

galactus' (and cyph1e's) solution is much simpler and much shorter, but I think separation of variables in this case is quite instructive:
$y' = -2(3y + 4)$

$dy = -2(3y + 4)~dx$

$\frac{dy}{3y + 4} = -2~dx$
(or you could keep the -2 with the y's. It makes no difference.)

$\int \frac{dy}{3y + 4} = -2 \int dx$

Now to integrate the left hand side.

Let $u = 3y + 4 \implies du = 3~dy$

So
$\int \frac{dy}{3y + 4} = \int \frac{du}{3u} = \frac{1}{3}ln|u| = \frac{1}{3}ln | 3y + 4 |$

So
$\int \frac{dy}{3y + 4} = -2 \int dx$

$\frac{1}{3}ln | 3y + 4 | = -2x + A$
(A is my undetermined constant.)

$ln | 3y + 4 | = -6x + 3A$

$| 3y + 4 | = e^{-6x + 3A}$

$| 3y + 4 | = e^{3A}e^{-6x}$

Now $e^{3A}$ is just a (positive) undetermined constant. Let's call this B.

$| 3y + 4 | = Be^{-6x}$

The way to deal with the absolute value bars is to split this into two equations:
$3y + 4 = Be^{-6x}$
and
$3y + 4 = -Be^{-6x}$

Since B is an undetermined (positive) constant anyway, let's say that we can have B be any constant (positive or negative) and just write this as one equation:
$3y + 4 = Ce^{-6x}$

$y = -\frac{4}{3} + Ce^{-6x}$

-Dan

**galactus** · July 12th 2008, 12:36 PM

but I think separation of variables in this case is quite instructive

That is what I thought. That is why I used the integrating factor instead of SOV. Apparently vodka didn't appreciate it, so I deleted it. Last time I answer a question by that poster.

**cyph1e** · July 12th 2008, 12:40 PM

Originally Posted by galactus

That is what I thought. That is why I used the integrating factor instead of SOV. Apparently vodka didn't appreciate it, so I deleted it. Last time I answer a question by that poster.

Even if he didn't appreciate it, please don't delete your post. It was a nice solution and other people viewing this thread might learn from it.

**galactus** · July 12th 2008, 12:43 PM

A bogaboo of mine is posters who thank one while shunning the other. It may be trifling, but that is how I see it.

**topsquark** · July 12th 2008, 02:11 PM

I have used my phenomenal cosmic powers to restore galactus' post. My reason is that other users can benefit from seeing the solution if they look up this thread. As I think galactus' solution was not only perfectly acceptable, but efficient and clearly explained I feel it should be included in this thread. (No offense to cyph1e's solution which I think is just as good.)

-Dan

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