Taylor Series Approximation Help

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Homework Help Overview

The discussion revolves around using Taylor series approximation to solve an initial value problem (IVP) involving differential equations. The original problem involves approximating values for a function defined by a differential equation, with a specific focus on the Taylor series method and programming aspects for computation.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the formulation of the Taylor series approximation and question the definitions of derivatives in the context of the problem. There are attempts to clarify the expressions for first derivatives and how they relate to the Taylor series. Some participants express uncertainty about the programming aspect of implementing the approximation.

Discussion Status

The discussion is active, with participants exploring different formulations and clarifying the problem's requirements. There is a focus on understanding the mathematical expressions involved and the challenges of translating these into a programming context. No consensus has been reached, and various interpretations of the problem are being examined.

Contextual Notes

Participants note changes in the initial value problem provided by the professor, which introduces new complexities. There is also mention of potential issues with the original problem's stability, prompting a reevaluation of the approach. The programming requirements are highlighted as a significant concern, with participants questioning the complexity of implementation.

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Homework Statement



Use the "Three Term" Taylor's approximation to find approximate values y_1 through y_20 with h=.1 for this Initial Value Problem:
y'= cosh(4x^2-2y^2)
y(0)=14

And write a computer program to do the grunt work approximation



Homework Equations





The Attempt at a Solution



y_n=cosh(4x^2-2y^2)
y''= (cosh(4x^2-2y^2)(4/3x^3-2/3y^3)
y_0=14
y_n+1=y_n+h{y_ncosh(4x_n^2-2y_n^2)+h^2/2[cosh(4x_n^2-2y_n^2)(4/3x_n^3-2/3y_n^3)]
I'm pretty sure I'm doing this wrong, and I also have no idea how to go about writing a computer program to do this work
 
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In the original problem statement, is that the second derivative, ie,

y'' = cosh(4x^2-2y^2)

or is it the first derivative?
 
wow, my professor changed the IVP to:
y'=xy^3 - cos(x)sin(y)
y(0)= -1
still using h= .1 and looking for y_1 to y_20, as well as writing the program. Apparently the original blew up too quickly.
 
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OK. Now to start, what were the various expressions that your prof gave you for approximations to first derivatives. Hint: a two term approx might be

[tex]y'_{+} = \frac{y_{n+1}-y_{n}}{h} + O(h)[/tex]

then again, it might be

[tex]y'_{-} = \frac{y_{n}-y_{n-1}}{h} + O(h)[/tex]

Do you remember how your prof established the three point approx?
 
Thanks for the help Theo, but my problem now lies in the programming bit of the problem. I think I've got the steps before it figured out. Please correct me if I'm wrong.

y''= x3(y^2)y' + y^3 -[(-sin(x)siny) + (cos(x)cos(y)y')]
= 3x(y^2)y' + y^3 +[cos(x)cos(y)y' - sin(x)sin(y)]
y'= (0)(-1)^3- [cos(0)sin(-1)] = sin(-1)
y''= 3(0) + (-1)^3 = [(cos(0)cos(-1)sin(-1)) - (sin(0)sin(-1))]
= -1 + sin(-1)
y_1= (x_0-y_0) = 0-(-1) = 1
y_2= y_1 + (x_1+y_1)h + (x_1+y_1+1)h^2/2
= 1 + (.1+1)(.1) + (.1+1+1).005 = 1.1205
y_3=y_2 + (x_2+y_2)h + (x_2+y_2+1)h^2/2
=1.1205 +(.2+1.1205)(.1) + (.2+1.1205+1).005 = 1.2641525
...

You see, now I have no idea how to go about writing a program for this approximation. What sort of program do I use, and where can I find it? Can I just download one? Is it very complicated to program this?
 
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