# 2nd Order Differential Equation with Improved Euler Method (Heun's)

## Homework Statement

I would like to solve a 2nd Order Differential Equation using the Improved Euler Method. The 2nd ODE is a Mass-Spring-Damper equation. I tried coming up with an solution for the Improved Euler Method, but not entirely sure. Can you help me and have a look if this is correct?
This solution assumes that inital conditions for x and u are known.

## Homework Equations

Forced Mass-Spring-Damper equation:

$m\ddot{x} + b\dot{x}+kx = F$

## The Attempt at a Solution

Converting the equation to a pair of first order differential equations:
$u=\dot{x}$
$\dot{u}=\ddot{x}$

And thus we have:
$\dot{u}=\frac{1}{m}\left[F-bu-kx\right]$

The forward Euler solution would result in:
$x_{n+1}=x_{n}+dt*u_{n}$
$u_{n+1}=u_{n}+dt*\dot{u_{n}}=u_{n}+dt*\frac{1}{m}\left[F_{n}-bu_{n}-kx_{n}\right]$

And the Improved Euler solution would be:
$x_{n+1}=x_{n}+dt*u_{n}$
$u_{n+1}=u_{n}+dt*\dot{u_{n}}=u_{n}+\frac{dt}{2} \left[ \frac{1}{m} \left[F_{n}-bu_{n}-kx_{n}\right]+\frac{1}{m}\left[F_{n+1}-bu_{n+1}-kx_{n+1}\right] \right]$

In the last equation, does the last $F$ needs to be $F_{n+1}$ like I have it?

Thanks a bunch, if you have time to have a look at it!

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