1. The problem statement, all variables and given/known data xy' - (x+1)y = 0, y(0) = 5 2. Relevant equations Well all of the differential equation stuff seems relevant to me. But from the beginning dy/dx + P(x)y = Q(x) and ye^{integral(P(x)dx)} = integral(Q(x)e^{integral(P(x)dx)}dx + C. The separation of variables M(x)dx + N(y)dy = 0. 3. The attempt at a solution Well I tried following separating everything out: xy' - (x+1)y = 0 y' - ((x+1)/x)y = 0/x then p(x) = -(x+1)/x and e^{integral(P(x)dx)} = e^{-integral((x+1)/x dx)} and using some more math we end up with e^{-x}/x and that is where I fall apart. Do I integrate this new function and then what? I come up with x/e^{-x} for y and (x-1)e^{x} for y' but when I put it back into the original equation i get x^{2} + x = x^{2} -x. Where did I go wrong?
Be careful...you're dividing by x but the initial condition is for x=0. You need to consider the function on an open interval about x=0 and so you cannot divide by x.