Differential equations/ separable

• Jul 28th 2010, 07:15 PM
Differential equations/ separable
I'm stuck on a question in my book, dy/dx = [y(1-y^2)]/[x(1-x^2)]
I know its separable but when i separate it, it comes to:

{1/[y(1-y^2)]} dy = {1/[x(1-x^2)]} dx and i dont know how to integrate both sides could someone please help
• Jul 28th 2010, 07:26 PM
Chris L T521
Quote:

I'm stuck on a question in my book, dy/dx = [y(1-y^2)]/[x(1-x^2)]
I know its separable but when i separate it, it comes to:

{1/[y(1-y^2)]} dy = {1/[x(1-x^2)]} dx and i dont know how to integrate both sides could someone please help

Observe that $\displaystyle \dfrac{1}{y(1-y^2)}=\dfrac{1}{y(1-y)(1+y)}$ (its the same for x as well...). Now apply partial fractions in order to integrate this.

Can you continue?
• Jul 28th 2010, 07:48 PM
we consider $\displaystyle \displaystyle\int\frac{dy}{y(1-y^2)}$ now if we put $\displaystyle y=\dfrac1t$ the integral becomes $\displaystyle \displaystyle\int\frac t{1-t^2}\,dt$ thus the original integral equals $\displaystyle \displaystyle\frac{1}{2}\ln \left( \frac{y^{2}}{y^{2}-1} \right)+k.$