This is a separable equation, which can be written as
\(\displaystyle{\left({2}{y}+{\cos{{y}}}\right)}{\left.{d}{y}\right.}={6}{x}^{{2}}{\left.{d}{x}\right.}\)

Integrate: \(\displaystyle∫{\left({2}{y}+{\cos{{y}}}\right)}{\left.{d}{y}\right.}=∫{6}{x}^{{2}}{\left.{d}{x}\right.}\)

LHS: \(\displaystyle∫{\left({2}{y}+{\cos{{y}}}\right)}{\left.{d}{y}\right.}={y}^{{2}}+{\sin{{y}}}+{C}{1},\) where C1 is some constant.

RHS: \(\displaystyle∫{6}{x}^{{2}}{\left.{d}{x}\right.}={2}{x}^{{3}}+{C}{2}\) where C2 is some constant.

So, from (1) it follows that \(\displaystyle{y}^{{2}}+{\sin{{y}}}={2}{x}^{{3}}+{C}\) where we defined C as C2-C1

Integrate: \(\displaystyle∫{\left({2}{y}+{\cos{{y}}}\right)}{\left.{d}{y}\right.}=∫{6}{x}^{{2}}{\left.{d}{x}\right.}\)

LHS: \(\displaystyle∫{\left({2}{y}+{\cos{{y}}}\right)}{\left.{d}{y}\right.}={y}^{{2}}+{\sin{{y}}}+{C}{1},\) where C1 is some constant.

RHS: \(\displaystyle∫{6}{x}^{{2}}{\left.{d}{x}\right.}={2}{x}^{{3}}+{C}{2}\) where C2 is some constant.

So, from (1) it follows that \(\displaystyle{y}^{{2}}+{\sin{{y}}}={2}{x}^{{3}}+{C}\) where we defined C as C2-C1