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 $$x(t + \tau) \approx x(t) + w_1 \tau f(x(t),t) + w_2 \tau f(x(t) + \beta \tau f(x(t),t), t + \alpha \tau)$$ 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 $$\frac{d^n}{dt^n}f(x(t),t) = \left(\frac{dx}{dt}\frac{\partial }{\partial x} + \frac{\partial }{\partial t}\right)^n f = \left(f\frac{\partial}{\partial x} + \frac{\partial}{\partial t}\right)^n f.$$

 

[jkthejetplane]

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

 

[pasmith]

$$\begin{align*} \left(f\frac{\partial}{\partial x} + \frac{\partial}{\partial t}\right)^2f &= \left(f\frac{\partial}{\partial x} + \frac{\partial}{\partial t}\right)\left(f\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t}\right) \\ & = f\frac{\partial}{\partial x}\left(f\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t}\right) + \frac{\partial}{\partial t}\left(f\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t}\right) \end{align*}$$ etc.