Rearranging separable equations

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In summary, the conversation discusses an equation and its solution steps. The equation is given as dy/dx = (y-4x)/(x-y) and it is rewritten as dy/dx = (y/x - 4)/(1-y/x) using the variable v=y/x. The conversation then moves on to differentiate the equation using the product rule, leading to the final equation dy/dx = v+x(dv/dx). The speaker expresses difficulty in understanding where the additional v term comes from, but is able to solve it with the help of another person.
  • #1
thedude36
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I'm having difficulty with the question in the pic provided. (http://i.imgur.com/Fg7CHoY.png). Basically the equation I am given is

dy/dx = (y-4x)/(x-y)​

and it walks through the steps needed to solve it, however, I am supposed to show how to arrive at each step given. I've gotten part a, where it asks to show that the above equation can be written as

dy/dx = (y/x - 4)/(1-y/x)​

and, introducing v=y/x, rewrite dy/dx in terms of v, x, and dv/dx. What I get for part b is

dy/dx = d(xv)/dx = x(dv/dx)​

however, on part c, it looks like I'm supposed to get

dy/dx = v+x(dv/dx)​

I'm not sure where the additional v term comes from. If anyone could help me out, i would really appreciate it!
 
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  • #2
thedude36 said:
dy/dx = d(xv)/dx = x(dv/dx)
dy/dx = v+x(dv/dx)​
It's just the product rule. How do you differentiate f(x)g(x)?
 
  • #3
I can't believe it was that simple - I had been staring at it for hours. Thank you!
 

1. What are separable equations?

Separable equations are first-order differential equations that can be rearranged into a form where all terms containing the dependent variable are on one side of the equation and all terms containing the independent variable are on the other side.

2. Why do we need to rearrange separable equations?

Rearranging separable equations makes it easier to solve them by isolating the dependent variable on one side of the equation. This allows us to use integration to find the solution.

3. How do you rearrange a separable equation?

To rearrange a separable equation, you need to move all terms containing the dependent variable to one side of the equation and all terms containing the independent variable to the other side. Then, you can integrate both sides to find the solution.

4. What are the steps for solving a separable equation?

The steps for solving a separable equation are: 1) rearrange the equation, 2) integrate both sides, 3) solve for the constant of integration, if necessary, and 4) substitute any known values to find the particular solution.

5. Are there any special cases in solving separable equations?

Yes, there are special cases such as when the dependent variable is not isolated on one side of the equation or when the integral on the other side cannot be evaluated. In these cases, additional techniques such as partial fractions or substitution may be necessary to solve the equation.

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