Find the general solution to the differential equation dy/dx = y/(x+1)(x+2)

1)Separating variables. Firstly, we need to get the y terms all on the dy/dx side and the x terms on the other side. In this case, we divide both sides by y which gives (1/y)dy/dx = 1/(x+1)(x+2).2) To make the RHS of the above equation easier to integrate, we need to turn the RHS into a partial fraction which will be of the form A/(x+1) +B/(x+2) so we can have the equality: 1/(x+1)(x+2) = A/(x+1) +B/(x+2). Multiplying both sides of this equation by (x+1)(x+2), we get 1 = A(x+2) + B(x+1). Substituting in x = -2 and x = -1, we can find the values of A and B which are 1 and -1 respectively. The resulting differential equation is (1/y)dy/dx = 1/(x+1) -1/(x+2).3) Integrating both sides with respect to x On the LHS, we get integral((1/y)dy/dx).dx. Now the dx s "cancel" so we integrate with respect to y and we get ln|y| on the LHS. On the RHS, we have integral (1/(x+1) - 1/(x+2)).dx which is ln|x+1| - ln|x+2|. Putting these together we get ln|y| = ln|x+1| - ln|x +2| + ln|k| where ln|k| is a constant. Using log laws , we can rearrange the RHS to getln|y| = ln|k(x+1)/(x+2)| . Finally, exponentiating both sides we get y = k(x+1)/(x+2).