Diff Eq - Should be easy, but my answer is upside down

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In summary, the conversation was about finding vz(r) with given boundary conditions and using the previous problem's equation. The solution was vz = W [ln (Ro / r) / ln(Ro/Ri)]. However, this was equivalent to the solution given by the answer key.
  • #1
Wildcat04
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Homework Statement



Find vz(r)

Boundary Conditions:

1. vz(Ro) = 0
2. vz(Ri) = W


Homework Equations



(1/r) d/dr [r dvz/dr] = 0 (from previous problem)

Let v = dvz / dr

d/dr [r v] = 0

r v = c1

v = c1 / r

vz = c1 ln r + c2

BC 1:

0 = c1 ln Ro + c2

c2 = - c1 ln Ro

BC 2:

W = c1 ln Ri - c1 ln Ro

W = c1 (ln Ri - ln Ro)

W = c1 ln(Ri/Ro)

c1 = W / ln(Ri/Ro)

My Soultion:

vz = (W ln r) / (ln(Ri/Ro) - (W ln Ro)/ln(Ri/Ro)

vz = W[ln (r / Ro) / ln(Ri/Ro)]


Unfortunately, the answer key says it should be:

vz = W [ln (Ro / r) / ln(Ro/Ri)]

So I am close but no cigar. I have recompleted this problem several times and keep arriving at the same solution. Can anyone point out my mistake?
 
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  • #2
Actually, the two answers are equivalent! Recall that
[tex]\frac{\ln\left(\frac{1}{a}\right)}{\ln\left(\frac{1}{b}\right)} = \frac{\ln\left(a^{-1}\right)}{\ln\left(b^{-1}\right)} = \frac{-\ln{a}}{-\ln{b}} = \frac{\ln{a}}{\ln{b}}[/tex]​
 
  • #3
Hrmmm...I always forget about identities, weither it be sin / cos or in this case natural logs.

Thank you foxjwill, I thought this was an easy one but I couldn't figure out how to get the correct answer, when, all along, I had it!
 

FAQ: Diff Eq - Should be easy, but my answer is upside down

What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is commonly used to model dynamic systems in various scientific fields, such as physics, engineering, and biology.

Why is solving differential equations important?

Solving differential equations is important because it allows us to predict the behavior of complex systems and make informed decisions. It is also the foundation for many advanced mathematical and scientific concepts.

What are the different types of differential equations?

There are three main types of differential equations: ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). ODEs involve a single independent variable, PDEs involve multiple independent variables, and SDEs involve random variables.

How do you solve a differential equation?

The method for solving a differential equation depends on its type and complexity. Some common methods include separation of variables, substitution, and using integrating factors. Advanced techniques such as Laplace transforms and numerical methods may also be used.

Can differential equations be solved by hand or do you need a computer?

Simple differential equations can be solved by hand using basic algebra and calculus. However, as the equations become more complex, it may be necessary to use a computer to find a solution. This is especially true for PDEs and SDEs, which often require numerical methods to find an approximate solution.

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