Question: In Problems 29–34, determine the Taylor series about the point x0 for the given functions and values of x0 .
The required expression is
For a function f(x) the Taylor series expansion about a point x0 is given by,
We have to calculate the Taylor series expansion for, at x0 =1 .
Calculating the derivatives of function at x0 .
f (x) =then f (x0) =1
f'(x) =then f'(x0) =
f''(x) = then f''(x0) =
f'''(x) = then f'''(x0) =
f''''(x) = then f''''(x0) =
Substituting the above derivatives in Taylor series expansion for the function at x0=1, then,
= role="math" localid="1664103079471"
Hence, the required expression is
Aging spring without damping. In a mass-spring system of aging spring discussed in Problem 30, assume that there is no damping (i.e., b=0), m=1 and k=1. To see the effect of aging consider as positive parameter.
(a) Redo Problem 30 with b=0 and η arbitrary but fixed.
(b) Set η =0 in the expansion obtained in part (a). Does this expansion agree with the expansion for the solution to the problem with η=0. [Hint: When η =0 the solution is x(t)=cos t].
To derive the general solutions given by equations (17)- (20) for the non-homogeneous equation (16), complete the following steps.
(a) Substitute and the Maclaurin series into equation (16) to obtain
(b) Equate the coefficients of like powers on both sides of the equation in part (a) and thereby deduce the equations
(c) Show that the relations in part (b) yield the general solution to (16) given in equations (17)-(20).
94% of StudySmarter users get better grades.Sign up for free