Euler's Method, Picard iteration: dy/dt = y - 3t, y(0) = 1

[khauna]

my assignment is to solve this initial value problem:

\(\displaystyle dy/dt=y-3t\) when \(\displaystyle y(0)=1\)

we are supposed to:
1.) solve the problem (i think you do with integrating factors)
2.) calculate three steps of Euler's method using delta t (step size) as 0.5
3.)calculate four more guesses using Picard iteration.

i really need help with 2 and 3 because I dont even know where to start...I been looking through a book but getting no where. Number 1 dont remember how to do integrating factors but I can figure that out but I will take any input.

All help is greatly appreciated, thanks.

~ Khauna

 

[Deleted member 4993]

my assignment is to solve this initial value problem:

\(\displaystyle dy/dt=y-3t\) when \(\displaystyle y(0)=1\)

we are supposed to:
1.) solve the problem (i think you do with integrating factors)
2.) calculate three steps of Euler's method using delta t (step size) as 0.5
3.)calculate four more guesses using Picard iteration.

i really need help with 2 and 3 because I dont even know where to start...I been looking through a book but getting no where. Number 1 dont remember how to do integrating factors but I can figure that out but I will take any input.

All help is greatly appreciated, thanks.

~ Khauna

Are you saying - your book does not show example for Euler's method of integration?

 

[galactus]

Using the integrating factor, we can rewrite:

\(\displaystyle \frac{dy}{dt}-y=-3t\)

The I.F. is \(\displaystyle e^{-\int dt}=e^{-t}\)

\(\displaystyle \frac{d}{dt}\left[ye^{-t}\right]=-3te^{-t}\)

Integrate:

\(\displaystyle ye^{-t}=3(t+1)e^{-t}+C\)

\(\displaystyle y=3(t+1)+Ce^{t}\)

Using the initial condition, y(0)=1:

\(\displaystyle 1=3(0+1)+Ce^{0}\Rightarrow C=-2\)

\(\displaystyle \boxed{y=3(t+1)-2e^{t}}\)

There, see how to do that now?. I showed you this so you can use it as a template, perhaps, on future problems involving I.F's

 

[khauna]

Thanks a lot.

No my book does have a section on Euler's method but I cant really follow what they are saying. I want an example of a problem done but the book for some reason doesn't have any examples...just a word explanation of how to do it. So I just need help on those or an example of something like those...

 

[Deleted member 4993]

Thanks a lot.

No my book does have a section on Euler's method but I cant really follow what they are saying. I want an example of a problem done but the book for some reason doesn't have any examples...just a word explanation of how to do it. So I just need help on those or an example of something like those...

You can use google to find several examples. Go to :

http://www.math.ubc.ca/~feldman/demos/demo1.html

or

http://www.esm.psu.edu/courses/emch12/i ... -tutorial/

If you study the theoretical part (and understand it) - you should be able to do this easily. Euler's method is the simplest of the numerical integartion methods. Study the graphical interpretation carefully.