# Which method for this DE?

1. May 11, 2004

### AndyW

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

$$\frac{dy}{dx}=\frac{4x-2y+4}{2x+y-2}$$

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?

2. May 11, 2004

### 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 $$x_{0},y_{0}$$ of the linear system:

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

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

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