Derive the first six terms of the taylor series about t 0

Question

A useful approach to analyzing a nonlinear equation is to study its linearized equation, which is obtained by replacing the nonlinear terms by linear approximations. For example, the nonlinear equation

d²θ/dt² + sinθ = 0,

which governs the motion of a simple pendulum, has

d²(θ)/dt² + θ = 0

as a linearization for small θ. (The nonlinear term sinθ has been replaced by the linear approxi­mation θ.)

A general solution to equation involves Jacobi elliptic functions, which are rather complicated, so let's try to approximate the solutions. For this purpose we consider two methods: Taylor series and linearization.

(a) Derive the first six terms of the Taylor series about t = 0 of the solution to equation with initial conditions 0(θ) = Π/12, θ'(0) = 0. 

(b) Solve equation subject to the same initial conditions θ(0) = Π/12, θ'(0) = 0.

(c) On the same coordinate axes, graph the two approximations found in parts (a) and (b).

(d) Discuss the advantages and disadvantages of the Taylor series method and the linearization method.

(e) Give a linearization for the initial value problem.

x''(t) + 0.1[ 1 - x²(t)]x'(t)] + x(t) = 0              x(0) = 0.4 ,    x'(0) = 0 ,

for x small. Solve linearized problem to obtain an approximation for the nonlinear problem.