Differential Equations: Separation of Variables.

Find the equation of the curve which satisfies the differential equation dy/dx=y/(x(x+1)) and passes through the point (1,2)

Your help is greatly appreciated!

Hello,

dy / dx = y / [x(x + 1)]
dy / y = dx / [x(x + 1)] = dx/x - dx/(x + 1)

Hence, by integrating separately:
ln|y| = ln|x| - ln|x + 1| + ln(C) with C the integration constant.
ln|y| = ln| Cx / (x + 1) |
y = Cx / (x + 1)

Since the curve passes through point (1; 2), we have:
2 = C/2
C = 4
y = 4x / (x + 1)

Check
y = 4x/(x + 1) = 4 - 4/(x + 1)
dy / dx = 4/(x + 1)² = [4x/(x + 1)] / [x(x + 1)] = y/[x(x + 1)]
QED

Hence the equation of the curve is:
y = 4x / (x + 1)

Logically,
Dragon.Jade :-)