Derivative Help: Find Taylor's Method of Order 3

[elle]

Derivative help please

Hi, can someone help me with the following question? The original question is to find Taylor's method of order 3 for the problem:

dy/dx = x/1+y

By following my notes I have worked out the answer for the first part of the question but I am having problems with the differentiation for the second derivative. Can anyone help?

Let f = x/1+y

df/dx = ( (1+y) - x dy/dx ) / (1+y)^2

substituting dy/dx by f:-

df/dx = ( (1+y) - x^2/(1+y)) / (1+y)^2

What would you get if you differentiate df/dx? I'm getting awfully confused with the many terms Do I use the quotient rule for this?

thanks for your time

 

[arildno]

No, this is not the way to go, you only complicate things!
I advise you to use separation of variables on this.
By the way, what do you mean by "Taylor's method"?


You did not mean just to compute the first few terms in the Taylor series, did you?

 

[elle]

Well I'm actually just starting a course on the numerical methods of ordinary differential equations and I've just come across a question to find the Taylor method of degree 3 for the differential equation.

I've just been following notes provided by the lecturer and following the examples.

So use separation of variables? How will I start it off? The only time I used this method was for finding the particular solution of a DE...

 

[arildno]

What do you mean by "Taylor method"??

Do you mean to find terms in the "Taylor series"?

 

[elle]

Yep find the terms.

 

[arildno]

And what is y(0)?

 

[elle]

The initial condition given is y(1) = 1

Here is the question:

Find Taylor's method of order 3 for the problem:

dy/dx = x/1+y,

y(1) = 1

 

[arildno]

So, what is y'(1)?
(Hint: What does your diff.eq tell you?)