# 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.  Compute the Taylor Series for about . [HINT: Multiply the series for by and
integrate.
2.  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.  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.  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.  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.

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

Recall these helpful expansions from calculus:

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