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...Chapter legendre s equation polynomials differential is one of the most important odes in physics it arises numerous problems particularly boundary value for spheres involves a parameter n whose depends on physical or engineering problem so actually whole family we solved example sec look back at any solution called function study these and other higher functions not occurring calculus theory special further will occur next sections dividing by obtain standard form needed theorem see that coefficients new are analytic may apply power series method substituting its derivatives into denoting constant simply writing first expression as two separate have help you to write out few terms each explicitly continue follows same general all four set thus instead three this gives note summation begins with since right side must be an identity if sum left zero now occurs fourth only third powers give brackets can written readily verify solving b well c formula recurrence relation recursion coeffic...