Not Understanding This Simplification

[snowJT]

I feel dumb because I can't see how you get from this:

$$x \frac{dy}{dx}+x^2y+y=0$$

to this

$$\frac{dy}{dx}+(x+\frac{1}{x})y=0$$

this is what I would of thought it would be...

$$\frac{dy}{dx}+(x+\frac{y}{x})+\frac{y}{x}=0$$

no?

 

[cristo]

Why do you think it should be $$\frac{dy}{dx}+(x+\frac{y}{x})+\frac{y}{x}=0$$?

Divide the original equation by x; what do you get?

 

[arildno]

Factorize $x^{2}y+y$ in the simplest manner.
What do you get?

 

[snowJT]

Why do you think it should be $$\frac{dy}{dx}+(x+\frac{y}{x})+\frac{y}{x}=0$$?

Divide the original equation by x; what do you get?

you get $$\frac{dy}{dx}+(\frac{x^2}{x}+\frac{y}{x})+\frac{y}{x}=0$$

Factorize $x^{2}y+y$ in the simplest manner.
What do you get?

you get $$\frac{x^2}{y}+1$$

 

[arildno]

No, you don't.

What does it mean to FACTORIZE?

 

[snowJT]

$$y(x^2+1)$$

oh I see now... thanks

but why are you not dividing the y that's on the most right side of the LHS by x?

 

[arildno]

So, you have now your diff.eq in the form:
$$x\frac{dy}{dx}+(x^{2}+1)y=0$$
Multiply this equation with 1/x; what do you get?

 

[snowJT]

sorry, I see it now... I'm just really bad at these things...