# Simple Taylor Series?

## Main Question or Discussion Point

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.
Thanks.

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HallsofIvy
Homework Helper
If, for example, you have a differential equation that says
$$\frac{d^2y}{dx^2}= f(x,y,\frac{dy}{dx})$$
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,
$$y(x)= 1- \frac{1}{2}x^2+ \frac{2}{4!}x^4$$

Of course much past there the derivatives are likely to become unwieldly.

Alright, thanks for that. :D

saltydog