Second oder ode solution with euler methods

𝟓𝒙̈+ 𝟕𝒙̇ + 𝟒𝒙 = 𝒔𝒊𝒏(𝟖𝒕) where, 𝒙(𝒕 = 𝟎) = 𝟓 and 𝒙̇(𝒕 = 𝟎) = 2 𝒙 values in the domain of [𝟎 𝟏𝟎]. with a step size of 𝚫𝒕 = 𝟎. 𝟓. How can I solve this system using euler methods ?

[James Tursa]

What have you done so far? What specific problems are you having with your code?

[Bayram FURKAN TORA]

I know solve first order ode but I don't know how to write script code to solve second ode

[Erivelton Gualter]

Hello Bayram,

You can easily use the ODE solvers from Matlab. Check the link bellow:

https://www.mathworks.com/help/symbolic/solve-a-single-differential-equation.html

Also, you can write your own method. Check the follow link:

http://tutorial.math.lamar.edu/Classes/DE/EulersMethod.aspx

Try to implement it and if you face a problem, share here your code and I will be glad to help.

[Bayram FURKAN TORA]

Thank you James

Answer 1

[James Tursa]

Rewrite your 2nd order equation as a pair of first order equations, then use Euler method on a 2-element vector. I.e.,

Define your 2-element state vector y as

y(1) is defined to be x
y(2) is defined to be xdot

The derivative of y(1) is y(2) by definition.

The derivative of y(2) can be found by solving your 2nd order DE for xdotdot.

See the van der Pol equation example in the doc here for an example of turning a 2nd order DE into a pair of 1st order DEs:

https://www.mathworks.com/help/matlab/ref/ode45.html?searchHighlight=ode45&s_tid=doc_srchtitle

You can essentially use your 1st order Euler code as an outline for this 2nd order system. Simply replace the scalar state with a 2-element vector state in your code.