Taylor Series Solution for y'' = e^y at x=0, y(0)=0, y'(0)=-1

In summary, the problem at hand is solving the differential equation y'' = e^y with initial conditions y(0)=0 and y'(0)=-1. The desired solution is the first six nonzero terms in the form of y(x) = y(0) + y'(0)x/1! + y''(0)x^2/2! + y'''(0)x^3/3! + ..., and the first two terms are known to be 0-x. The challenge lies in finding the y''(0) term, which can be obtained by substituting u=y' and using the fact that y'''=(y'')'=d/dx(e^y)=y'e^y. The
  • #1
dduardo
Staff Emeritus
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This problem has been bugging me and I can't seem to figure it out:

y'' = e^y where y(0)= 0 and y'(0)= -1

I'm supposed to get the first 6 nonzero terms

I know the form is:

y(x) = y(0) + y'(0)x/1! + y''(0)x^2/2! + y'''(0)x^3/3! +...

and I know the first two terms are

y(x) = 0 - x +...

But what's troubling is the e^y. How would I go about getting the y''(0) term. I tried a subsitution of u = y', but the integrals gets very messy. I'm thinking there is either a typo in the book or there is a simpler way to get the answer.

All I need is an example of how to get the next term. I can figure out the rest.
 
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  • #2
y''(0) = e^{y(0)} = e^0=1

if you don't mind I'll also say that y''' = (y'')' = d/dx(e^y) = y'e^y
 
  • #3
Ah, ok, thanks alot. I understand now.
 

What is a Taylor Series?

A Taylor Series is a mathematical representation of a function as an infinite sum of terms, with each term being a multiple of a particular derivative of the function at a specific point.

What is the purpose of a Taylor Series?

The purpose of a Taylor Series is to approximate a function with a polynomial that can be easily manipulated and calculated, making it useful in solving complex mathematical problems.

How is a Taylor Series different from a Maclaurin Series?

A Maclaurin Series is a special case of a Taylor Series, where the function is approximated around the point x=0. A Taylor Series can be centered at any point on the function's domain.

What is the formula for a Taylor Series?

The general formula for a Taylor Series is: f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ..., where a is the center of the series and f'(a), f''(a), f'''(a), etc. are the derivatives of f(x) evaluated at a.

How is a Taylor Series used to approximate functions?

A Taylor Series can be used to approximate functions by taking a finite number of terms in the series and using them to calculate an approximate value for the function at a specific point. As more terms are added to the series, the approximation becomes more accurate.

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