Given Differential equations is
\(\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}={1}+{y}^{{2}}\)
with y(0)=0. Now
PSKdy/dx=1+y^2 -> dy/(1+y^2)=dx
->∫(dy/1+y^2)=∫dx+CZSK, where C is a integrating constant
\(\displaystyle\to{\arctan{{y}}}={x}+{C}\)
Now, \(\displaystyle{y}{\left({0}\right)}={\left({0}\right)}\to{C}={0}\)
Hence, the solution is
arctany=x->y=tanx