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Chapter 8: Q2E (page 450)
In Problems 1-10, use a substitution y=xr to find the general solution to the given equation for x>0.
2x2y"(x)+13xy'(x)+15y(x)=0
The general solution to the given equation 2x2y"(x)+13xy'(x)+15y(x)=0 is y=c1x-5/2+c2x-3.
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Duffing's Equation. In the study of a nonlinear spring with periodic forcing, the following equation arises:
Letand.Find the first three nonzero terms in the Taylor polynomial approximations to the solution with initial values.
Use the change of variables \(s = \frac{2}{\alpha }\sqrt {\frac{k}{m}} {e^{ - \alpha t/2}}\)to show that the differential equation of the aging spring \(mx'' + k{e^{ - \alpha t}}x = 0\),\(\alpha > 0\)becomes\({s^2}\frac{{{d^2}x}}{{d{s^2}}} + s\frac{{dx}}{{ds}} + {s^2}x = 0\).
Question: In Problems 29鈥�34, determine the Taylor series about the point X0for the given functions and values of X0.
32. f(x)=ln(1+x), x0 =0
To derive the general solutions given by equations (17)- (20)for the non-homogeneous equation (16), complete the following steps.
(a) Substituteand 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).
Question: To find the first few terms in the power series for the quotient q(x) in Problem 15, treat the power series in the numerator and denominator as "long polynomials" and carry out long division. That is, perform
16.
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