Finding the M.G.F of product of two random variables.

Careless mistake at second last line: \begin{align} \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{\dfrac{-t^2y^2}{2}}e^{\frac{-y^2}{2}}dy &= \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{\dfrac{-(t^2+1)y^2}{2}}dy\\ &=\frac{1}{\sqrt{t^2+1}} \end{align}

Edit:

There is actually a mistake earlier. Thanks, tmrlvi for pointing out.

In the $4^{th}$ line as we complete the square: \begin{align} & \dfrac{1}{2\pi}\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{-\dfrac{(x-ty)^2}{2}}e^{\dfrac{t^2y^2}{2}}e^{\frac{-y^2}{2}}dxdy \\ &= \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{\dfrac{t^2y^2}{2}}e^{\frac{-y^2}{2}}dy \\ &= \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{-\frac{y^{2}(1-t^2)}2}dy \\ &= \dfrac{1}{\sqrt{1-t^2}} \end{align}