Question

Determine the first three nonzero terms in the Taylor polynomial

series approximations for the following IVP. y"+siny=0 y(0)=1,
y'(0)=0

Answer 1

The first three nonzero terms in the Taylor polynomial series approximation for the given IVP are y₁ = 1, y₂ = 0, y₃ = 0.

Step-by-step explanation:

The Taylor polynomial series approximations for the given initial value problem (IVP) can be found by using the formula for the Taylor polynomial:

y(x) = y(c) + y'(c)(x - c) + y''(c)(x - c)^2/2! + ...

Substituting the given initial conditions y(0) = 1 and y'(0) = 0 into the equation, we can find the first three nonzero terms in the Taylor polynomial series approximation. The terms are:

y₁ = 1, y₂ = (0)(x - 0) + (0)(x - 0)^2/2! = 0, y₃ = (0)(x - 0)^2/2! + (0)(x - 0)^3/3! = 0