First, read the introduction to Exercise. Consider the general Cauchy −Euler equationExerc...

First, read the introduction to Exercise. Consider the general Cauchy −Euler equation

Exercise

It was stated in the Closure thatany given Cauchy - Euler equation can be reduced to a constant - coefficient equation by the change of variables x= et. In this exercise we ask you to try that idea for some specific cases; in the next exercise we ask for a general proof of the italicized claim. Lety(x(t))= Y(t), and lety′andY′denotedy/dxanddY/dt,respectively.

(xnDn+ c1xn−1Dn −1+ … + cn−1 xD +cn) y= 0,

whereD=d/dx.Letx=et,and definey(x(t)) = Y(t).

The results (8.2) suggest that the formula

xkDky = D (D− 1) … (D − k +1)Y,(8.3)

holds for all positive integersk.Prove that (8.3) is indeed correct. HINT: Use mathematical induction. That is, assume that (8.3) holds for any given positivekand, by differentiating both sides with respect tox,show that

xk+1 Dk+1 = D(D − 1) … (D − 1) y(8.4)

which is the same as (8.3) but withkchanged tok +1. Thus,

it must be true that (8.3) holds for all positive integersk.