# Numerical Analysis/Order of RK methods/Implicit RK2 on an Autonomous ODE

We consider an autonomous initial value ODE**String Module Error: function rep expects a number as second parameter, received "**

**"****(**)**String Module Error: function rep expects a number as second parameter, received "**

**"**

Applying the Tradezoidal rule

gives the implicit Runge-Kutta method**String Module Error: function rep expects a number as second parameter, received "**

**"****(**)**String Module Error: function rep expects a number as second parameter, received "**

**"**

We will show that (**method**
) is second order.

Expanding the true solution about using Taylor series, we have

Since satisfies (**ODE**

), we can substitute and obtain**String Module Error: function rep expects a number as second parameter, received "**

**"****(**)**String Module Error: function rep expects a number as second parameter, received "**

**"**

In (**method**
) we can assume since that is the previous data.
Subtracting (**method**
) from (**true**

) gives us the local truncation error**String Module Error: function rep expects a number as second parameter, received "**

**"****(**)**String Module Error: function rep expects a number as second parameter, received "**

**"**

In order to cancel more terms we need to expand .
However, so we cannot do a regular Taylor expansion.
Instead we can plug (**method**

) back into and then do a Taylor expansion to obtain**String Module Error: function rep expects a number as second parameter, received "**

**"****(**)**String Module Error: function rep expects a number as second parameter, received "**

**"**

Substituting (**implicit**
) into (**error1**

) yields**String Module Error: function rep expects a number as second parameter, received "**

**"****(**)**String Module Error: function rep expects a number as second parameter, received "**

**"**

This substitution was productive since the terms canceled.
We can do this trick again, but this time only need (**implicit**
) up to since everything will be multiplied by at least and this can go into the .
Substituting (**implicit**
) in for the first occurance of in (**error2**

) yields**String Module Error: function rep expects a number as second parameter, received "**

**"****(**)**String Module Error: function rep expects a number as second parameter, received "**

**"**

This substitution was productive since the terms canceled. We can do this again, now truncating (**implicit**
) at .
Substituting (**implicit**
) into (**error3**
) yields

Since the term does not cancel, we have shown that the local truncation error is and thus the method is order 2.