Numerical Methods: Euler to Runge–Kutta · Hard problem

The second-order Taylor method advances by $y_{n+1} = y_n + h\,y'(t_n) + \dfrac{h^2}{2}\,y''(t_n)$, where $y''$ is found by differentiating the equation. Apply two steps with $h = \dfrac{1}{2}$ to $y' = ty$, $y(0) = 1$, to estimate $y(1)$. Express your answer as a common fraction.