Solving 2tv' - v = 0: Is it Separable?

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In summary, the problem is finding a solution for 2tv' - v = 0, which is stated to be separable. The individual is still learning and would appreciate extra details. The solution involves rearranging the equation and integrating both sides, resulting in v = k*sqrt(t). To verify this solution, one can take the derivative and confirm that 2tv' - v = 0.
  • #1
seand
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How do you find a solution for:

2tv' - v = 0

The text says it's separable but I'm not seeing it. I'm just learning so extra details are appreciated. Thanks.

(this should have been posted in the homework section - but I can't seem to move it there, sorry)
 
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  • #2
2t dv/dt = v
==> 2t dv = v dt
Can you continue?
 
  • #3
well I think I want to switch things around so v and t are separated

1/v dv = 1/2t dt

integrating both sides
ln(v) = 1/2 ln(t)+C

e^x both sides:
v = k*sqrt(t)

Which looks right! Is that how I was meant to do it? If so, thanks - I got stalled before when I got the ln() on both sides.
 
  • #4
Looking right might not be good enough. You can check by taking the derivative and verifying that tv' - v = 0.
 
  • #5
Mark44 said:
Looking right might not be good enough. You can check by taking the derivative and verifying that tv' - v = 0.
Verifying that 2tv' - v = 0 :smile:
 

Related to Solving 2tv' - v = 0: Is it Separable?

1. What does it mean for an equation to be "separable"?

When an equation is "separable," it means that it can be written in the form of two separate equations, each containing only one variable. For example, in the equation 2tv - v = 0, we can separate out the t and v variables into two equations: 2t = 0 and v = 0. This makes it easier to solve for each variable individually.

2. How do I know if an equation is separable?

An equation is separable if it can be rewritten in the form of two separate equations, each containing only one variable. In the equation 2tv - v = 0, we can separate out the t and v variables into two equations: 2t = 0 and v = 0. If you are unsure, it is always helpful to try and separate out the variables and see if you end up with two separate equations.

3. What are the benefits of solving a separable equation?

Solving a separable equation makes it easier to find a solution for each individual variable. This can be beneficial when dealing with more complex equations, as it allows us to break down the problem into smaller, more manageable parts. It also gives us a better understanding of how each variable is related to the overall equation.

4. Can all equations be made separable?

No, not all equations can be made separable. Some equations are inherently more complex and cannot be broken down into two separate equations with only one variable each. In these cases, other methods of solving may be necessary.

5. How do I solve a separable equation?

To solve a separable equation, you first need to separate out the variables into two separate equations, each containing only one variable. Then, you can solve each equation individually to find the value of each variable. Finally, you can combine the solutions to get the final solution for the original equation. It is important to check your solutions and make sure they work in the original equation.

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