Which method for this DE?

[AndyW]

I'm new to DEs, and can do most I've come across so far, but this has me stumped:

[tex]\frac{dy}{dx}=\frac{4x-2y+4}{2x+y-2}[/tex]

Out of these methods which should I use?

1 Integrating Factor
2 Seperable
3 Bernoulli
4 Change of Variable (y=xv)

I'm pretty sure it's the 4th but I've not been able to find a solution. Could someone confirm that this is the correct approach?

 

[arildno]

Basically, yes, however:
You'll remain stumped as long as you do not "eliminate" the constants in your expression (that would be +4 in the numerator, -2 in the denominator).

In order to eliminate these constants in an acceptable manner, use the following trick:
Find the solutions [tex]x_{0},y_{0}[/tex] of the linear system:

4x-2y+4=0
2x+y-2=0

This yields: [tex]x_{0}=0,y_{0}=2[/tex]
Now change the scales:
[tex]\hat{x}=x-x_{0}, \hat{y}(\hat{x})=y(x)-y_{0}[/tex]

In the hatted variables, you now have the differential equation:
[tex]\frac{d\hat{y}}{d\hat{x}}=\frac{4\hat{x}-2\hat{y}}{2\hat{x}+\hat{y}}[/tex]

This should help you solve the problem..

 

[M98Ranger]

It is indeed the correct approach to use the change of variable (y=xv) method for this differential equation. This method is used when the equation is not in a standard form that can be easily solved by other methods such as separation of variables or Bernoulli's method. By substituting y=xv, the equation can be transformed into a separable form and then solved using the standard methods.

However, it is important to note that not all differential equations can be solved by a single method. Sometimes, a combination of methods may be needed to find a solution. So, it is always a good idea to try different methods and see which one works best for a particular equation. In this case, it seems like the change of variable method is the most suitable one, but if you are unable to find a solution using that method, you can try other methods as well.