I'm having some trouble solving the differential equation $$\ \frac{dy}{dx}= \cos^2\left(\frac{\pi y}{2}\right)$$ when $y = 0.5$ its $x=0$ and I need to find the value of x when $y=2.5$, anyone able to help?
2026-04-24 23:09:11.1777072151
Differential equation with cosine squared
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Just separate variables.
Remember that $(\tan u)' = \sec^2 u$, so $(\tan (\pi u/2))' = \pi/2 \sec^2 (\pi u/2)$.
Using this, $\int \frac{dy}{\cos^2(\pi y/2)} = \int \sec^2(\pi y/2) dy = \frac{2}{\pi} \tan ( \pi y/2)$.
So, you have $\int \frac{dy}{\cos^2(\pi y/2)} = \frac{2}{\pi} \tan ( \pi y/2) = \int dx + C = x + C$. Now plug in your intial conditions and solve for C.