Help with 2nd order Runge Kutta and series expansion

[jkthejetplane]

Homework Statement

2nd order Runge Kutta problem

Relevant Equations

Second order Runge Kutta

So here's my homework question:

This is the reference formula along with the Rung-Kutta form with the variables mentioned in the question

Here is my attempt so far:

Problem is that i am unsure how to expand this to even get going. I tried referencing my text Math Methods by Boas which has a lot on series and expansion but I am kinda stumped. Even after i get it expanded I am not entirely sure where to go with it
Any help? This class is killing me haha

Thanks in advance

 

[pasmith]

You need to expand x(t + \tau) to third order in \tau, and compare that to what you get from expanding <br /> x(t + \tau) \approx x(t) + w_1 \tau f(x(t),t) + w_2 \tau f(x(t) + \beta \tau f(x(t),t), t + \alpha \tau) to third order in \tau (which means that f(x(t) + \beta \tau f(x(t),t), t + \alpha \tau) only has to be expanded to second order, as it is already multiplied by \tau).

Remember that by definition \frac{dx}{dt}(t) = f(x(t),t) and that by the chain rule <br /> \frac{d^n}{dt^n}f(x(t),t) = \left(\frac{dx}{dt}\frac{\partial }{\partial x} + \frac{\partial }{\partial t}\right)^n f<br /> = \left(f\frac{\partial}{\partial x} + \frac{\partial}{\partial t}\right)^n f.

 

[jkthejetplane]

What does eq 3.25 have to do with it? and i have no idea how to expand either of those

 

[pasmith]

\begin{align*}<br /> \left(f\frac{\partial}{\partial x} + \frac{\partial}{\partial t}\right)^2f &amp;=<br /> \left(f\frac{\partial}{\partial x} + \frac{\partial}{\partial t}\right)\left(f\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t}\right) \\<br /> &amp; = f\frac{\partial}{\partial x}\left(f\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t}\right) + \frac{\partial}{\partial t}\left(f\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t}\right)<br /> \end{align*} etc.