The solution of the differential equation `(dy)/(dx)=1+x+y^2+x y^2,\ y(0)=0` is a. `y^2=e x p(x+(x^2)/2)-1` b. `y^2=1+C \ exp(x+(x^2)/2)` c. `y="tan (C + x + x^2)` d `y=tan(x+(x^2)/2)`

Explanation

`frac{d y}{d x}=1+x+y^{2}+x y^{2} `

`Rightarrow frac{d y}{d x}=(x+1)+y^{2}(x+1) `

`Rightarrow frac{d y}{d x}=(x+1)(1+y^{2}) `

`Rightarrow frac{d y}{(1+y^{2})}=(x+1) d x`

Integrating both sides

`int frac{d y}{(1+y^{2})}=int(x+1) d x `

`Rightarrow tan ^{-1} y=frac{x^{2}}{2}+x+C ldots . .(1)`

`y(0)=0 `

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