Question

Find the general solution to the differential equation y´ = sec^2 (πt).

Answer 1

\[\frac{dy}{dt}=\sec^2\pi t\\
\int dy=\int\sec^2\pi t~dt\\
y=\frac{1}{\pi}\tan\pi t+C\]

Answer 2

If you don't see the equivalence right away between the integral and the general solution, consider a substitution:
\[u=\pi t~~\Rightarrow~~du=\pi~dt~~\iff~~\frac{1}{\pi}du=dt\]
so then
\[\int \sec^2\pi t~dt=\frac{1}{\pi}\int\sec^2u~du=\frac{1}{\pi}\tan u+C=\frac{1}{\pi}\tan\pi t+C\]

Answer 3

Thank you that helped a lot!