Mathematics - Differential Equation Question with Solution | TestHub

$\frac{dy}{dx}=25y^2-4$
So$\frac{dy}{25y^2-4}=dx$
Integrating$\frac{1}{25}\times \frac{1}{2\times \frac{2}{5}}\ln \left|\frac{y-\frac{2}{5}}{y+\frac{2}{5}}\right|=x+c$
$\Rightarrow \ln \left|\frac{5y-2}{5y+2}\right|=20\left(x+c\right)$
Now,$c=0$as$f\left(0\right)=0$
Hence,$\left|\frac{5y-2}{5y+2}\right|=e^{\left(20x\right)}$
$\underset{x\to -\infty }{\lim }\left|\frac{5f\left(x\right)-2 }{5f\left(x\right)+2}\right|= \underset{x\to -\infty }{\lim }{e}^{\left(20x\right)}$
Now,
$RHS=0\Rightarrow \underset{x\to -\infty }{\lim }\left(5f\left(x\right)-2\right)=0$
$\Rightarrow $ \underset{x\to -\infty }{\lim }f\left(x\right)=\frac{2}{5}$$