## A treatise on differential equations: and on the calculus of finite differences |

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arbitrary constant arbitrary functions Assume ux auxiliary equation Calculus complementary function complete integral consequently constant coefficients continued fraction convergent denoted determine differential coefficients divergent dx dy dy dx eliminated equa equal equation of differences equivalent exact differential expression factor find a curve find the integral Finite Differences formula fraction given gives Hence homogeneous function imaginary roots increment independent variables infinite involving Lagrange's Theorem Let ux linear equation method multiplied obtained partial differential equations particular integral particular values primitive equation proposed equation proposed series quantity reduced result satisfies the proposed second member second order separating the symbols shews Similarly singular solution substituting successive values suppose symbols of operation tangent Taylor's theorem Theorem tion uxvx value of ux variables are separated zero