Finding Complementary Function | Partial Differential Equation | SNS Institutions

#snsinstitutions #snsdesignthinkers #designthinking In solving linear partial differential equations with constant coefficients, the complete solution is expressed as: General Solution = Complementary Function (C.F.) + Particular Integral (P.I.) . General Solution=Complementary Function (C.F.) + Particular Integral (P.I.). The Complementary Function (C.F.) is obtained by solving the corresponding homogeneous equation, i.e., the equation obtained when the right-hand side is zero. When solving linear partial differential equations with constant coefficients, the complete solution is made up of two parts: the complementary function and the particular integral. The complementary function is obtained from the homogeneous part of the equation, that is, the form of the equation where the right-hand side is taken as zero. To find the complementary function, an auxiliary relation is formed by replacing the differential operators with symbolic constants. The solutions of this auxiliary relation determine the form of the complementary function. Each solution corresponds to a family of arbitrary functions that appear in the complementary function. If the auxiliary relation has distinct solutions, the complementary function is expressed as a combination of arbitrary functions, each depending on a linear relation between the variables. If the auxiliary relation has repeated solutions, additional factors involving the independent variables are included along with the arbitrary functions. Thus, the complementary function represents the general solution of the homogeneous equation, and it provides the essential structure upon which the complete solution of the given equation is built.