# Determining the first three nonzero terms for Taylor polynomial approximation for IVP

• Dec 3rd 2011, 03:37 PM
s3a
Determining the first three nonzero terms for Taylor polynomial approximation for IVP
I only know how to do this with regular equations. How do I do this with a differential equation involved?

Any help would be greatly appreciated!
• Dec 3rd 2011, 05:07 PM
TKHunny
Re: Determining the first three nonzero terms for Taylor polynomial approximation for
You have not understood the definition.

The first term requires y(0), whci is zero. It is therefore not a non-zero term.

The second term is related to 8*sin(0) + 7*e^(0) = 7 -- This should lead you to the first non-zero term.

The third term whould be related to y".

You need values of the function and its derivatives. When they give you a derivative, don't be afraid of it.
• Dec 3rd 2011, 07:03 PM
s3a
Re: Determining the first three nonzero terms for Taylor polynomial approximation for
Treating y as a constant, I get:

y'' = 7e^x = 7e^x

Yielding y(x) = 7x + 7x^2/2! + 7 x^3/3! but that is incorrect. I am thinking I shouldn't be treating y as constant but I don't know what else to do.
• Dec 3rd 2011, 07:29 PM
chisigma
Re: Determining the first three nonzero terms for Taylor polynomial approximation for
Quote:

Originally Posted by s3a
I only know how to do this with regular equations. How do I do this with a differential equation involved?

Any help would be greatly appreciated!

Let's write again the DE...

$\displaystyle y^{'}= 8\ \sin y + 7\ e^{x},\ y(0)=0$ (1)

Under the assumption that the solution is analytic in x=0, we can write...

$\displaystyle y(x)= \sum_{n=0}^{\infty} \frac{y^{(n)}(0)}{n!}\ x^{n}= \sum_{n=0}^{\infty} a_{n}\ x^{n}$ (2)

The derivatives of y in x=0 are computed as follows...

$\displaystyle y(0)=0 \implies a_{0}=0$

... and from (1)...

$\displaystyle y^{'}(0)= 7 \implies a_{1}=7$

Now we compute $\displaystyle y^{''}$ from (1)...

$\displaystyle y^{''}= 8\ \cos y\ y^{'} + 7\ e^{x}$ (3)

... and from (3)...

$\displaystyle y^{''}(0)= 8 \cdot 7 +7 = 63 \implies a_{2}= \frac{63}{2}$

Now You can proceed to compute succesive terms... if necessary...
• Dec 4th 2011, 02:34 AM
TKHunny
Re: Determining the first three nonzero terms for Taylor polynomial approximation for
Quote:

Originally Posted by s3a
Treating y as a constant,

Why would you do that? y is a function of x. Please note the chain rule as demonstrated by chisigma.