Here the angular acceleration is constant, so we have,

$\theta =\omega t+\frac{1}{2}\alpha t^2$.

When flywheel is rotated from rest in the first second, then ${\theta }_1=\frac{1}{2}\alpha {\left(1\right)}^2=5 \mathrm{rad}$

$\Rightarrow \alpha =10 \mathrm{rad} {\mathrm{s}}^{-2}$

The angle turned by flywheel in the next second is ${\theta }_2=\frac{1}{2}\alpha {\left(2\right)}^2=\frac{1}{2}\left(10\right)\left(4\right)=20 \mathrm{rad} {\mathrm{s}}^{-2}$

Thus, the angle turned during next second is $\theta =20-5=15 \mathrm{rad}$