Implicit and explicit solution for a given initial-value problem

1. Sep 25, 2011

Cheruby

The equation is:
x2 dy/dx= y - xy
IC (initial conditions): y(-1) = -1 (This is used to solve for C)
Must first separate the variables x and y and then integrate them and solve for y, but I got stuck...

x2dy = (y - xy)dx
x2dy = y(1-x)dx
dy= y(1-x)dx/x2 <-- not sure what to do with the y now... I figure I could divide everything by y so I could bring it to the left side but the right side would have dx/y

The answer is y = e-(1+1/x)/x

Last edited: Sep 25, 2011
2. Sep 25, 2011

Jackx

You are doing it right, but when you divide by Y you will have dy/y, and that is perfectly acceptable. The right side you will have (1-x)/x^2 dx. Then integrate both sides.

3. Sep 25, 2011

Cheruby

But wouldn't the right side have dx/y? That's why i'm stuck, I'm not sure if it's acceptable!

4. Sep 25, 2011

Jackx

dx can be treated as just another factor on the right side. So since they are all factors when you divide by y the y just goes away on the right side.

Now if it was y + (whatever)dx, then yes if you divided by y you would have a dx/y.

5. Sep 25, 2011

Cheruby

Alright thank you Jackx