A three-stage explicit Runge-Kutta method for solving the autonomous ordinary differential equation

$$\frac{d y}{d t}=f(y)$$

is given by

$$y_{n+1}=y_{n}+h\left(b_{1} k_{1}+b_{2} k_{2}+b_{3} k_{3}\right),$$

where

$$\begin{aligned} &k_{1}=f\left(y_{n}\right) \\ &k_{2}=f\left(y_{n}+h a_{1} k_{1}\right) \\ &k_{3}=f\left(y_{n}+h\left(a_{2} k_{1}+a_{3} k_{2}\right)\right) \end{aligned}$$

and $h>0$ is the time-step. Derive sufficient conditions on the coefficients $b_{1}, b_{2}, b_{3}, a_{1}$, $a_{2}$ and $a_{3}$ for the method to be of third order.

Assuming that these conditions hold, verify that $-\frac{5}{2}$ belongs to the linear stability domain of the method.