How Do You Solve This First Order ODE with Given Condition?

[mkay123321]

The question is x^2dy/dx + y^2=0 , y(1)=3

I re-arrange the equation to get -1/y^2dy=1/x^2dx

separated them, then I integrate both sides to get 1/y=-1/x + c

Now I don't get how they got the answer y=3x/(4x-3), as when I try use the condition I get a different answer, could anyone help? I might have done something wrong in the integration?

 

[TylerH]

You just solve it for y: $$y=\frac{-x}{Cx+1}$$. Then set y(1) equal to 3 (since that's the condition you're given) and solve (using good ol' algebra) for C: $$y(1)=\frac{-1}{C+1}=3$$
So, C=-4/3 and after you plug C into y, it is what your book says it is.