differential equation: x^2 dy/dx + xy = 1

[Clifford]

x^2 dy/dx + xy = 1

I know I have to get all the y's on one side with the dy and the x's on the other with dx, but I can't seem to rearrange this.

my attempt:

x^2dy + xydx = dx
x(xdy + ydx) = dx
xdy + ydx = dx / x
xdy = dx(1/x - y)
xdy/dx=1/x - y

Kind of seems like I am going around in a circle with this problem.

 

[galactus]

You need to find an integrating factor in this one, besides just separating variables.

\(\displaystyle x^{2}\frac{dy}{dx}+xy=1\)

Divide by x^2 and get:

\(\displaystyle \frac{dy}{dx}+\frac{1}{x}y=\frac{1}{x^{2}}\)

Multiply by the integrating factor, which is \(\displaystyle e^{\int\frac{1}{x}dx}=e^{ln(x)}=x\):

You get \(\displaystyle \frac{d}{dx}[xy]=\frac{1}{x}\)

Now, integrate and you get:

\(\displaystyle xy=ln(x)+C\)

\(\displaystyle \boxed{y=\frac{ln(x)}{x}+\frac{C}{x}}\)