Prove that the third-order Runge-Kutta method reproduces the Taylor series of the solution up to and including terms inh3for any differential equations.
$$x(t+h) =x(t) + \frac19 (2 K_1 + 3 K_2 + 4 K_3)$$
where
$$\begin{aligned} K_1 &= h f(t,x)\\ K_2 &= h f \left(t + \frac12 h, x + \frac12 K_1 \right)\\ K_3 &= h f \left(t + \frac34 h, x + \frac34 K_2 \right)\end{aligned}$$
I am really struggling with proving this. I need any help for that. Thank you.
The Butcher tableau for the given method is $$ \begin{array}{c|c}c&A\\\hline&b\end{array} \qquad=\qquad \begin{array}{c|ccc} 0&\\ \frac12&\frac12\\ \frac34&0&\frac34\\ \hline &\frac29&\frac39&\frac49 \end{array} $$ For 3-stage methods you get order conditions (see for instance Butcher 2008) \begin{align} \text{order $1$}&:& b_1+b_2+b_3&=1\\ \text{order $2$}&:& b_2c_2+b_3c_3&=\tfrac12\\ \text{order $3$}&:& b_2c_2^2+b_3c_3^2&=\tfrac13\\ & & b_3a_{32}c_2&=\tfrac16\\ \end{align}
It can be easily checked that these conditions are satisfied.