Find Taylor Polynomial of a degree of 4 f x6y3 and f03Solut

Solution

dy/dx =6y-3
integrating both sides
dy/(6y-3) =dx
ln(6y-3)=x+c
6y-3=e^(x+c)

(putting condition at x=0 y=3)

15=e^c

=>c=2.7

=>y=(15*e^x+3)/6

f¹(x)=2.5 *e^x

f¹¹(x)=2.5 *e^x, f¹¹¹(x)=2.5 *e^x, f¹¹¹¹(x)=2.5 *e^x

writing taylor polynomial

= f(0) +f¹(0)*x/1! + f¹¹(0)*x^2/2! +........

=3+2.5x+2.5*(x^2)/2 +2.5*(x^3)/6 +2.5*(x^4)/24