Question

Express 2x/ x²-1 in partial fractions and hence find the general solution of the differential equation dy/dx = 2xy / x² - 1 expressing y explicitly in terms of x.

Answer 1

The given function $f(x)=\frac{2x}{x^2-1}$

$\frac{dy}{dx}=\frac{2xy}{x^2-1}$

We can write it as,

$\frac{dy}{y}=\frac{2xdx}{x^2-1}$

let $x^2-1= t$

$2xdx=dt$

Hence,

$\frac{dy}{y}=\frac{dt}{t}$

Now, taking the integration of the above equation,

$\int \frac{dy}{y}=\int \frac{dt}{t}$

$\Rightarrow \ln y=\ln t +c$

Now, substituting the value of t

$\ln y = \ln (x^2-1)+c$

Or, we can write it as,

$\Rightarrow \ln y -\ln(x^2-1)=c$

$\Rightarrow \ln\frac{y}{x^2-1}=c$

$\Rightarrow \frac{y}{x^2-1}=e^c$