5 Find the first six nonzero terms of a Taylor series soluti

5. Find the first six nonzero terms of a Taylor series solution centered at 0 of the given IVP problem, y\" = xy?-y, y(0)= 2, y, (0) = 1

Solution

Solution: given y₂=xy²-y_{1............................}1)                     y(0)=2 & y₁(0)=1

the tayler series at centered 0

y(x)=y(0)+y₁(0)*x+y₂(0)/2*(x²)+y₃(0)/6*(x³)+y₄(0)/24*(x⁴)+y₅(0)/120*(x⁵)                         upto six terms

from eq. no. 1) diffrentiate both side and using the given value, to get y₂=-1

as like thies step apply we get y₃=1, y₄=7, y₅=-5    putting in teylor series

y(x)=2+x-x²/2+x³/6+7x⁴/24-5x⁵/120                          upto six terms