Taylors approximation problem

Jul 2010
10
0
dx/dt = 1- sin[ ln(1+x )]

Replace the right hand side by the linear Taylor's approximation about x0=0 and solve the resulting equation. Determine the particular solution for the initial condition x(0)=0.
 

Jester

MHF Helper
Dec 2008
2,470
1,255
Conway AR
Taylor series for \(\displaystyle \ln(1+x)\) and \(\displaystyle \sin x\)

\(\displaystyle \ln (1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} \cdots\)
\(\displaystyle \sin x = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} \cdots\)

so to first order the taylor series would be

\(\displaystyle \sin (\ln(1+x)) = x\)
 
Jul 2010
10
0
Hadn't learnt the Taylor series so had to do a quick crash course, this helped a lot, thanks heaps.