7.2.5-ODEs: Implicit Euler's Method

These videos were created to accompany a university course, Numerical Methods for Engineers, taught Spring 2013. The text used in the course was "Numerical Methods for Engineers, 6th ed." by Steven Chapra and Raymond Canale.

[MichaelTheThinker]

I needed you to show me a problem involving implicit euler method. Oddly, my textbook skipped over giving an example for this particular method. And saying it’s a variation of the explicit euler method doesn’t help me.

[MrAntiKnowledge]

Well that didnt help me at all.
How can I calculate *y_i+1* if it's dependant on some function I have to pop in *y_i+1*?

Nevermind, figured it out. For those who had similar problems an example:
f(x, y(x))= y ' = a * y(x) ( a is just some constant value)
=>
y_i+1 = y_i + h * f(x_i+1, y_i+1) = y_i + h * a * y_i+1
y_i+1 - h * a * y_i+1 = y_i
y_i+1 ( 1 - h * a ) = y_i
y_i+1 = y_i / ( 1 - h * a )

Basically, fill in the formula and throw the terms around until you got y_i+1 on one side only, then you can use your y_0 for y_i and start calculating away.
In my example you could even get rid of y_i since its always divided by a constant value(I doubt that's always the case) =>
y_i+1 = y_0 / ( 1 - h * a )^(i+1).
(sorry for the shitty format)

[Mulkek]

Awesome video & explain so clearly!

[MechanicalEI]

Jacob Bishop, a very good introduction to Implicit Euler Method. You writing down the encourages me to make notes too. Thanks for uploading!

[SuperMaDBrothers]

So to explain the explicit Euler -- I mean the implicit Euler method