# Taylors approximation problem

#### zcus05

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
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

$$\displaystyle \sin (\ln(1+x)) = x$$

#### zcus05

Hadn't learnt the Taylor series so had to do a quick crash course, this helped a lot, thanks heaps.