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absolute term adjoint algebraic ascending powers binomial coefficients Chapter Clairaut's equation cofactors common factor complementary function consider constant multiple convenient convergence corresponding course definite integrals descending series disposable constants elementary hypergeometric eliminate equation of order Euler's identity exact derivative example expansion finite given homogeneous hypergeometric differential equation hypergeometric equation hypergeometric series indicial equation infinite series integrating factor interval Legendre's equation Lemma limit linear differential equation linear equation logarithmic solution method negative integer notation number of disposable obtain operator pair parameters particular integral particular solution Pn(x polynomial of degree polynomial solution positive integer precisely principal integral prove quadratures recurrence formula reduced equation replace right-hand member roots satisfies the differential second solution Show Similarly simple solution singular solution Solve the equations substitution sufficiently suppose terminated series theorem theory values vanishes variable write written Wronskian zero