How Do You Solve Integration Issues with Differential Equations?

[PPapadopoulos]

Hello All,

I am trying to solve the following problem and I am having some trouble in doing so...would it be possible for someone to help me?

Here it is:

(-dP/dZ)*(r/2)=k((dv/dr)^n)

(-dP/dZ) is constant and I am trying to integrate dv/dr but I am having trouble separating the term... please HELP!

 

[Mute]

So, your problem is

$$\frac{\alpha r}{2} = k \left(\frac{dv}{dr} \right)^n$$

where $\alpha = -\frac{dP}{dz}$. Take the n^th root of both sides to get

$$\frac{dv}{dr} = \left(\frac{\alpha r}{2 k}\right)^n$$

which is easily seen to be separable. Just be careful, though - taking the n^th root could result in numerous solutions, e.g. n = even results in a +/-, so you may need to choose just one of the signs based on the context of the problem, and I'm assuming everything should be real, so that's the only issue with taking the root. (n odd is okay, since there's only one real root).

 

[PPapadopoulos]

If you take the nth root of both sides do you not end up with (ar/2k)^1/n?

 

[PPapadopoulos]

I am sorry let me post the complete equation: do you not get dv/dr=(ar/2k)^1/n

 

[Mute]

Ah, sorry. 'Twas a typo or some sort of brain misfire when typing. Yes, you get that, so you can separate it as

$$\frac{dr}{r^{1/n}} = \left(\frac{\alpha}{2k} \right)^{1/n}dv$$