Question

Find the general solution of the differential equation dy/dx = xy / (x+2)(x+1) expressing y explicitly in terms of x.

Answer 1

$\frac{{dy}}{{dx}} = \frac{{xy}}{{(x + 2)(x + 1)}} \Rightarrow\\ \frac{{dy}}{y} = \frac{{xdx}}{{(x + 2)(x + 1)}} \Rightarrow \frac{{dy}}{y} = \frac{{2x + 2 - x - 2}}{{(x + 2)(x + 1)}}dx \Rightarrow \\\frac{{dy}}{y} = \left( {\frac{{2(x + 1)}}{{(x + 2)(x + 1)}} - \frac{{x + 2}}{{(x + 2)(x + 1)}}} \right)dx \Rightarrow \frac{{dy}}{y} = \left( {\frac{2}{{x + 2}} - \frac{1}{{x + 1}}} \right)dx \\\Rightarrow \ln y = 2\ln (x + 2) - \ln (x + 1) + \ln C \Rightarrow \ln y = \ln \frac{{C{{(x + 2)}^2}}}{{x + 1}} \Rightarrow y = \frac{{C{{(x + 2)}^2}}}{{x + 1}}$

Answer: $y = \frac{{C{{(x + 2)}^2}}}{{x + 1}}$