# Manipulation of Arbitrary Constants in Differential Equations

1. Mar 6, 2013

### Bill Nye Tho

1. The problem statement, all variables and given/known data

yy''+(y')2 = 0

2. Relevant equations

yv(dv/dy)+v2=0

3. The attempt at a solution

Variable separable when solving for the first step the result is:

- ln |v| = ln |y| + C1

Now, I've looked at the remainder of the solution with a few other sources and the cause of my mistake results in the constant.

After turning the equation into: ln |vy| + C1 = 0;

I raise everything to the e so that I can solve for v.

It seems all the solutions do that as well but yet they don't raise the constant to the e.

Is it because eC1 will still be a constant and therefore completely arbitrary?

2. Mar 6, 2013

### SammyS

Staff Emeritus
$\displaystyle e^{C_{\,1}}\$ is positive, otherwise it's as arbitrary as using a redefined C1 .

3. Mar 6, 2013

### HallsofIvy

Staff Emeritus
You could, at this point, take the exponential of both sides:
$e^{-ln|v|}= e^{ln|y|+ C_1}$
$\frac{1}{|v|}= e^{C_1}|y|$
Now, as you say, since C1 is an arbitrary constant, so is $e^{C_1}$ so let's just call it "C2". And if allow C2 to be either positive or negative, it we can drop the absolute values: $\frac{1}{v}= C_2y$ which is, of course, the same as $yv= \frac{1}{C_2}$ or $yv= C_3$ where $C_3= 1/C_2$.
Some texts just don't bother to label the different constants differently.

4. Mar 6, 2013

### Bill Nye Tho

That makes sense, thank you.

Seems like they just don't want things to get messy so they reuse the same constants.

5. Mar 6, 2013

### HallsofIvy

Staff Emeritus
A "conservation of constants" law!!!