(1) Seek power series solutions of the given differential equation about the given point&nbsp;x0, find the recurrence relation.(2) Find the first tour terms in each of two solutions&nbsp;y1&nbsp;and&nbsp;y2&nbsp;(unless the series terminates sooner).(3) By evaluating the Wronskian&nbsp;W(y1y2)(x0), show that&nbsp;y1&nbsp;and&nbsp;y2&nbsp;form a fundamental set of solutions.(4) It possible, find the general term in each solution&nbsp;(1+x2)y4xy′+6y=0,x0=0

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(1) Seek power series solutions of the given differential equation about the given point&amp;nbsp;x0, find the recurrence relation.(2) Find the first tour terms in each of two solutions&amp;nbsp;y1&amp;nbsp;and&amp;nbsp;y2&amp;nbsp;(unless the series terminates sooner).(3) By evaluating the Wronskian&amp;nbsp;W(y1y2)(x0), show that&amp;nbsp;y1&amp;nbsp;and&amp;nbsp;y2&amp;nbsp;form a fundamental set of solutions.(4) It possible, find the general term in each solution&amp;nbsp;(1+x2)y4xy′+6y=0,x0=0

Laith Petty · Accepted Answer

1) For this equation&amp;nbsp;x0&amp;nbsp;is an ordinary point so we look for a soultion in the form of a power series about&amp;nbsp;x0y=∑n=0+∞anxnwhich converges in some interval&amp;nbsp;|x|&amp;lt;p&amp;nbsp;The series for y and y&quot; are given byy′=∑n=1+∞nanxn−1,&amp;nbsp;y∑n=2+∞n(n−1)anxn−2Consequently, by changing the initial differential equation by plugging these equalities in, we are able to(1+x2)=∑n=2+∞n(n−1)anxn−2−4x∑n=1+∞nanxn−1+6∑n=0+∞anxn=0⇔∑n=2+∞n(n−1)anxn−2+∑n=2+∞n(n−1)anxn−4∑n=1+∞nanxn+6∑n=0+∞anxn=0⇔∑n=0+∞(n+2)(n+1)an+2xn+∑n=2+∞n(n−1)anxn−4a1x−4∑n=2+∞nanxn+6a0+6a1x+6∑n=2+∞anxn=0&amp;nbsp;

(a) Seek power series solutions of the given differential equation about the giv

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