Answer to Question #223119 in Discrete Mathematics for Iron rose

Question

Express 2x+1 / x(x+1) in partial fractions and hence find the general solution of the differential equation x(x+1)dy/dx = y(2x+1) expressing y explicitly in terms of x.

Answer 1

It follows that $\frac{2x+1}{x(x+1)}=\frac{1}{x}+\frac{1}{x+1}$ is the expression in partial fractions. Let us find the general solution of the differential equation $x(x+1)\frac{dy}{dx} = y(2x+1)$ expressing $y$ explicitly in terms of $x$. We get that $\frac{dy}{y} = \frac{2x+1}{x(x+1)}dx,$ and hence $\int\frac{dy}{y} = \int\frac{2x+1}{x(x+1)}dx=\int(\frac{1}{x}+\frac{1}{x+1})dx.$ We conclude that $\ln|y|=\ln|x|+\ln|x+1|+\ln|C|=\ln|x(x+1)C|.$ Therefore, $y=Cx(x+1)$ is the general solution of the differential equation $x(x+1)\frac{dy}{dx} = y(2x+1)$.