Suppose the spring is heated to that the spring “constant

| August 31, 2017

Question
As a spring is heated, its spring “constant” decreases (see the figure).

Suppose the spring is heated to that the spring “constant” at time t
is given by k(t)=6-t (N/m). If the unforced mass-spring system
has a mass m of two kg with a damping constant of b=1 (N-sec-/m)
with initial conditions x(0)=3 and x^’ (0)=0, then the displacement
x(t) is governed by the initial value problem:

2x”(t)+x'(t)+(6-t)x(t)=0, x(0)=3, x'(0)=0.

Find at least the first four nonzero terms in the power series expansion about t=0 for the displacement.

Math 246 Homework Assignment Six
Do the following problems showing all your work. (Point values are given in the square
brackets.)
1. [4] Compute the Taylor Series for about . [HINT: Multiply the series for by and
integrate.
2. [4] Determine at least the first four non-zero terms in the power series expansion
about x=0 for the general solution of the differential equation:

3. [4] Use the first few terms of the power series expansion to find a cubic
polynomial approximation for the solution of the initial value problem:

Graph the linear, quadratic, and cubic polynomial approximations on the interval [-5, 5].
4. [5] As a spring is heated, its spring “constant” decreases (see the figure).
Suppose the spring is heated to that the spring “constant” at time t
is given by (N/m). If the unforced mass-spring system
has a mass m of two kg with a damping constant of b=1 (N-sec-/m)
with initial conditions and , then the displacement
is governed by the initial value problem:

Find at least the first four nonzero terms in the power series expansion about t=0 for the
displacement.
5. [5] The Legendre equation of order is the second order differential equation:

This differential equation appears in a variety of applications ranging from
numerical integration (for example, Gaussian quadrature) to the problem of
determining the steady-state temperature of a spherical ball (given the temperature
on the boundary). Use the series substitution and derive a recursive relation on
the coefficients . (Hint: you should be able to rewrite in terms of a multiple of )
6.

[3] Rewrite the given equation as a first order system of the matrix form .

Recall these helpful expansions from calculus:

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