Linear Differential Equation: when x=x(y)

[rygza]

(x+y^2)dy=ydx
rewrote as: dx/dy - x/y = y
Realized I had P(y)x and Q(y) rather than the P(x) and Q(x) from equations where y is a function of x. My problem now is after I multiply by the Integrating factor (-1/y):

-1 - x/(y^2) + 1/y(dx/dy)

I tried to make exact but i don't know the proper variables to use. I used

(partial derivative. M/partial deriv. x) = - 1/(y^2) = (partial N/partial y)

Is this proper? Usually for y=y(x) functions it's (partial M/partial y) but if i use that for this problem it doesn't make the equation exact. OR Am I supposed to rewrite so i have dy/dx (and then use the y=y(x) method)?

 

[jackmell]

How about if I write it as:

$$xdy+y^2dy=ydx$$

$$xdy-ydx=-y^2dy$$

Ok, we know:

$$d\left(\frac{y}{x}\right)=\frac{xdy-ydx}{x^2}$$

So that left side could be written as:

$$x^2d\left(\frac{y}{x}\right)=-y^2dy$$

Now what happens if I divide throughout by y^2?

 

[rygza]

How about if I write it as:

$$xdy+y^2dy=ydx$$

$$xdy-ydx=-y^2dy$$

Ok, we know:

$$d\left(\frac{y}{x}\right)=\frac{xdy-ydx}{x^2}$$

So that left side could be written as:

$$x^2d\left(\frac{y}{x}\right)=-y^2dy$$

Now what happens if I divide throughout by y^2?

lost me on the d(y/x) = ... part

 

[tiny-tim]

lost me on the d(y/x) = ... part

hi rygza!

i suppose you're happy with d(xy) = xdy + ydx ?

that's the equivalent of (xy)' = xy' + yx'.

ok, now start with (y/x)' = (xy' - yx')/x²,

and you get d(y/x) = (xdy - ydx)/x².

 

[blue_raver22]

I would suggest that you consult with a mathematics expert or refer to a textbook on differential equations for a more accurate and comprehensive response to your question. However, based on my understanding of differential equations, your approach seems correct. Since the equation is in terms of y, it would be appropriate to use the partial derivative with respect to y. However, if this does not lead to an exact equation, you may need to rewrite the equation in terms of x and use the y=y(x) method. It is also possible that there may be other methods or techniques that could be applied to solve this particular differential equation. It would be best to consult with a mathematics expert for a more in-depth analysis and solution.