Please show full steps Solve y yt Sec y 1 t2 with y 0 0 Solu

Please show full steps

Solution

Solution:

Given y\' =   e^{(y - t)} (1 + t²) sec(y)

It means

dy/dt = e^{(y - t)} (1 + t²) sec(y)

dy/dt = e^y/e^t (1 + t²) sec(y)

dy = e^y/e^t (1 + t²) sec(y) dt

dy / (e^y) = 1/e^t (1 + t²) sec(y) dt

cos(y)/(e^y) dy = (1 + t²)/e^t dt

- 1/2 e^{- y} (cosy - siny) = 2( - e ^-t t - e^{- t}) - e^-t (t² + 1) + C