One more diff equation. Do i still use integrating method to differentiate this one?
$\displaystyle \frac{dy}{dt} = \frac{1 + y^2}{y(1 + t^2)}$ is seperable:
$\displaystyle \frac{y}{1 + y^2} \, dy = \frac{1}{1 + t^2} \, dt$
$\displaystyle \Rightarrow \int \frac{y}{1 + y^2} \, dy = \int \frac{1}{1 + t^2} \, dt$
$\displaystyle \Rightarrow \frac{1}{2} \ln (1 + y^2) = \tan^{-1} (t) + C$
and you can make y the subject if you want ......
Two small mistakes I think:
1. $\displaystyle y = \sqrt{e^{2 (\tan^{-1} t + C)} - 1}$. The square root goes over the 1 as well.
2. It's probably easier to sub t = 1, y = 5 into $\displaystyle \frac{1}{2} \ln (1 + y^2) = \tan^{-1} (t) + C$:
$\displaystyle \frac{1}{2} \ln (1 + 25) = \tan^{-1} (1) + C$
$\displaystyle \Rightarrow \ln(26) = \frac{\pi}{2} + 2C$ etc.