Solving Differential Equations with Simple Taylor Series Method

In summary, a method for solving differential equations of 1st (And maybe 2nd?) order can be found by using a simple Taylor series.
  • #1
Pseudo Statistic
391
6
Hi,
I was reading this math book once... and it had a method for solving differential equations of 1st (And maybe 2nd? I don't remember) order by using simple Taylor series...
I didn't even have to understand much of what was going on, except that I followed some simple rule and I ended up with an accurate solution.
Anyone know where I can find more information about this method? All of the other ones I'm finding show power series which look fairly tedious to compute.
Thanks.
 
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  • #2
If, for example, you have a differential equation that says
[tex]\frac{d^2y}{dx^2}= f(x,y,\frac{dy}{dx})[/tex]
with initial conditions y(a)= A, y'(a)= B, you can use the equation directly to find y"(a), differentiate the equation to get a formula for y"'(x) and then evaluate to get y"'(a), differentiate again, etc. so that you can get as many derivatives, evaluated at x= a, as you want and construct the Taylor Polynomial. Especially nice for non-linear equations.

Specific example: y"= x2- y2 with initial conditions y(0)= 1, y'(0)= 0.

Immediately y"(0)= 02-(12)= -1.

y"'= 2x- 2yy' so y"'(0)= 2(0)- 2(1)(0)= 0.

yiv= 2- 2(y')2- 2yy" so
yiv(0)= 2- 2(02)-2(1)(0)= 2.

yv= -4(y')2- 2y'y"- 2yy"' so
yv(0)= -4(02)- 2(0)(-1)- 2(1)(0)= 0

To fifth order,
[tex] y(x)= 1- \frac{1}{2}x^2+ \frac{2}{4!}x^4[/tex]

Of course much past there the derivatives are likely to become unwieldly.
 
  • #3
Alright, thanks for that. :D
 
  • #4
Thanks, may work with this a bit. You guys mind? You know, solve the DE numerically, then calculate the Taylor series as Hall suggests, then plot the two and see how they match. Hey Pseudo, why don't you do that, say for the equation Hall used. You can take the easy approach like I would do: Have Mathematica calculate the derivatives and just string them together. :smile:
 
  • #5
Hahah...
If I had Mathematica, that would probably be what I would do... ;)
I'm still looking into solving these numerically with the improved Euler's method... but I guess I'm a little too impatient to be using numerical methods forever.. :P
Looks like Taylor Series is a little short-cut to solving specific equations, heh.
 

1. What is a Simple Taylor Series?

A Simple Taylor Series is a mathematical tool used to approximate a function by breaking it down into a polynomial of infinite terms. It is based on the idea that any function can be represented by an infinite sum of derivatives at a single point.

2. How is a Simple Taylor Series calculated?

To calculate a Simple Taylor Series, we need to know the function's derivatives at a specific point. These derivatives are then used to construct the polynomial, with each derivative having a coefficient determined by the function's value at that point. The more terms included in the series, the more accurate the approximation will be.

3. What is the purpose of a Simple Taylor Series?

The purpose of a Simple Taylor Series is to approximate a function with a simpler polynomial expression. This can be useful in situations where the original function is complex or difficult to work with, but the polynomial expression is easier to manipulate and analyze.

4. What are the limitations of a Simple Taylor Series?

A Simple Taylor Series can only approximate a function within a certain radius of convergence, which is determined by the function's behavior at the point of expansion. If the function has discontinuities or singularities within this radius, the series will not be able to accurately represent it.

5. How is a Simple Taylor Series different from a Taylor Series?

A Simple Taylor Series only includes the first derivative of a function, while a Taylor Series includes all derivatives. This means that a Simple Taylor Series is a special case of a Taylor Series, with a simpler form and more limited scope. Additionally, a Taylor Series can be centered at any point, while a Simple Taylor Series is centered at a specific point.

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