Factorising and integrating a differential

In summary, the conversation discusses a problem with an equation involving δ2V/δr2 + 1/r δV/δr - V/r = 0 and finding the equation for V explicitly. The correct equation is determined to be V = C1r/2 + C2/r and the solution involves Bessel functions. A substitution of r=exp(y) is recommended and it is noted that the equation is now a homogeneous ODE and can be solved using the classical method.
  • #1
eddysd
39
0
Having a bit of trouble with this equation, I need to find V explicitly and this would obviously be done by factorising and integrating, but I can't seem to factorise it correctly. I have what I think is the correct answer but can't do the steps to get there. Any help would be greatly appreciated.

δ2V/δr2 + 1/r δV/δr - V/r = 0

which I think goes to:

V = C1r/2 + C2/r

C1 and C2 being constants of integration.
 
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  • #2
hi eddysd! :smile:
eddysd said:
δ2V/δr2 + 1/r δV/δr - V/r = 0

which I think goes to:

V = C1r/2 + C2/r

no, i think that would be δ2V/δr2 + 1/r δV/δr - V/r2 = 0
 
  • #3
tiny-tim said:
hi eddysd! :smile:


no, i think that would be δ2V/δr2 + 1/r δV/δr - V/r2 = 0

Yes sorry, typo, that is the correct equation! Do you think you could tell me how to get from that to the equation for V?
 
  • #4
eddysd said:
δ2V/δr2 + 1/r δV/δr - V/r = 0
Solutions involves Bessel functions.
 
  • #5
substiution r=exp(y) will give you the required answer .To know why multiply the eqn by r^2 and see the second term and it does not involve bessel functions.
 
Last edited:
  • #6
eddysd said:
Yes sorry, typo, that is the correct equation! Do you think you could tell me how to get from that to the equation for V?

Now it is an homogeneous ODE.
So, use the classical method : Let V=r*W and rewite the ODE where W is the unknown function.
 

1. What is factorising a differential?

Factorising a differential is the process of breaking down a differential equation into simpler components. This is done by finding common factors in the equation and factoring them out. The goal is to make the equation easier to solve.

2. Why is factorising important in solving differentials?

Factorising is important in solving differentials because it allows us to simplify the equation and make it easier to solve. It also helps us to identify any repeating patterns or relationships within the equation, which can lead to more efficient solutions.

3. What is the purpose of integrating a differential?

Integrating a differential is the process of finding the antiderivative of a given function. This allows us to find the original function from its derivative, which can be useful in a variety of applications, such as calculating areas under a curve or solving differential equations.

4. How do you integrate a differential?

The process of integrating a differential involves finding the antiderivative of the function, which is the inverse operation of differentiation. This can be done using various techniques, such as integration by substitution, integration by parts, or using specific integration formulas.

5. What are some common mistakes to avoid when factorising and integrating differentials?

Some common mistakes to avoid when factorising and integrating differentials include not properly identifying and factoring out common factors, making errors in the integration process, and forgetting to add the constant of integration. It is important to double-check all steps and formulas used in the process to ensure accurate solutions.

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