hey dudes i need some help if possible as i'm really stuck on this question and cant find out how to work it out anywhere!
a) if a function y=f(x) satisfies the ODE
dy/dx= F(x,y)
(so that f ' (x)= F(x, f(x)), for all x). show that you can obtain a taylor series for the function f around the point x=a, where f(a)=y0
(you do not need to justify the necessary differentiability conditions of f or F).
use the idea to show if :
dy/dx = x + y^2, y(1)=1, y ' (1)=2,
then (under certain suitable conditions)
y(x)=1+2(x-1) + (x-1)^2 + 5/6(x-1)^3+...
(you do not need to find any other terms in this series).
PLEASE ANYONE IF YOU DO KNOW HOW TO WORK THIS OUT PLEASE PLEASE PLEASE HELP ME AS I AM REALLY STUCK ON HOW TO WORK THIS OUT!!
THANK YOU
a) if a function y=f(x) satisfies the ODE
dy/dx= F(x,y)
(so that f ' (x)= F(x, f(x)), for all x). show that you can obtain a taylor series for the function f around the point x=a, where f(a)=y0
(you do not need to justify the necessary differentiability conditions of f or F).
use the idea to show if :
dy/dx = x + y^2, y(1)=1, y ' (1)=2,
then (under certain suitable conditions)
y(x)=1+2(x-1) + (x-1)^2 + 5/6(x-1)^3+...
(you do not need to find any other terms in this series).
PLEASE ANYONE IF YOU DO KNOW HOW TO WORK THIS OUT PLEASE PLEASE PLEASE HELP ME AS I AM REALLY STUCK ON HOW TO WORK THIS OUT!!
THANK YOU