Show that the differential equation of which `y=2(x^2-1)+c e^-x^2` is a solution, is `(dy)/(dx)+2x y=4x^3dot`

Explanation

`y=2(x^{2}-1)+c e^{-x^{2}}`

`
frac{d y}{d x}=2(2 x)+c e^{-x^{2}} times-2 x`

`
frac{d y}{d x}=4 x-2 x c e^{-x^{2}}...(i) `

`
y=2(x^{2}-1)+c e^{-x^{2}}`

`
c e^{-x^{2}}=y-2 x^{2}+2 ldots (ii) `


using the ce e^{-x^{wedge} 2} from eqn. (ii) to eqn. (i),

`
frac{d y}{d x}=4 x-2 x(y-2 x^{2}+2) `

`
frac{d y}{d x}=4 x-2 x y+4 x^{3}-4 x`

`
...