Taylor Series Approximation Help

[tiptop]

Hi everybody, I was given this assignment, but I honestly have no idea how to go about solving it, and we are also to write some sort of program that does the work for us, and have a hard copy of it available to turn in. I don't have the slightest idea where to begin in that area either. I don't know why he expects us to know how to do that... unless its remarkably easy. So anyway I hope I can get some help here.

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

Could someone help me out with this mindbender, I don't even know how to approach it with the hyperbolic cos. And if you can you provide me with the program, or at least tell what I have to use, and how to use it?

Thanks a bunch!

[HallsofIvy]

Hi everybody, I was given this assignment, but I honestly have no idea how to go about solving it, and we are also to write some sort of program that does the work for us, and have a hard copy of it available to turn in. I don't have the slightest idea where to begin in that area either. I don't know why he expects us to know how to do that... unless its remarkably easy. So anyway I hope I can get some help here.

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

Could someone help me out with this mindbender, I don't even know how to approach it with the hyperbolic cos. And if you can you provide me with the program, or at least tell what I have to use, and how to use it?

Thanks a bunch!
I notice that you have posted this same thing on a number of different math forums- and haven't shown that you have actually made any attempt at it yourself!

Perhaps your professor expected that, since you are taking differential equations, you would KNOW what a Taylor's polynomial was! It is fairly common topic in Calculus. You also should have seen cosh(x) and sinh(x) in calculus.

The "three term" (second order) Taylor's polynomial for a function f, about x= 0, is given by f"(0)/2)x^2+ f'(0)x+ f(0). You are told that f(0)= 14. you are told that f'(x)= cosh(4x^2-2y^2) and f'(0)= cosh(4(0)^2- 2(14)^2)= cosh(392).
Differentiating f'(x)= cosh(4x^2-2y^2) with respect to x gives
f"(x)= (cosh(4x^2- 2y^2))'= (8x- 2y y') sinh(4x^2- 2y^2)= (8(0)- 2(14)(cosh(392))(sinh(4(0)^2- 2(14)^2)
Now you just need to write a program that will loop from i= 1 to 20, taking x= i(0.1) and calculate Ax^2+ Bx+ C where A, B, and C are as calculated above.

[tiptop]

Yes I understand that, but the problem is I don't know how to write a program for this, is there some program that I need to download in order to write this?

Yeah, and my professor changed the IVP to be
y'=xy^3 - cos(x)sin(y)
y(0)= -1
with h= .1

[HallsofIvy]

Perhaps you need to ask the professor about the "pre-requisites" for this course! He can't very well insist that a person write a program as a requirement for the course if he has not already required that the person know HOW to write a program! You haven't said what course this is. If this were, for example, a "Numerical Analysis" course, being able to write at least simple programs would be expected to be a pre-requisite.

[tiptop]

The class is Calc 2, and the assignment is extra credit, so I guess we're not required to understand programming, but he's not offering any help at all.