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PFAFFIAN Differential Equations And It's Solutions

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PFAFFIAN Differential Equations And It's Solutions :-      The relation              n             Σ  Fᵢ(x₁,x₂,......xₙ) dxᵢ = 0            i=1   where Fᵢ's (i=1,2,....n) are functions of all or some of the n independent variables x₁,x₂,.....xₙ is called a Pfaffian differential equation . Here we shall study Pfaffian differential equation in three independent variables which is of the form              P dx + Q dy + R dz = 0    ...........(1)    where P, Q , R are functions of x , y and z . Setting -->                        -->               X = (P,Q,R)  and dr = (dx,dy,dz), equation (1) can be written in the vector notation as            ...
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Complete Integral Of Partial Differential Equations

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Complete Solution Or Complete Integral Of Partial Differential Equations :   Definition :              A solution of partial differential equation is said to be a complete solution or complete integral if it contains as many arbitrary constants as there are independent variables . Definition Of General Solution Or Integral :        A general solution or integral of a partial differential equation is a relation involving arbitrary functions which provides a solution to that equation . Linear Partial Differential Equation Of The First Order :        A partial differential equation of first order is said to be linear if it is of the first degree in P and Q otherwise it is            non linear . Example :   (i) Linear Partial Differential                    ...
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Conditions Of Integrability

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Conditions Of Integrability :    ( A ) First Form Of Integrability :                   A necessary and sufficient condition for the integrability of a bounded function f is that to every ε > 0 , there corresponds δ > 0 such that for every partition P of [a,b] with norms μ(P) < δ ,   U(P,f) - L(P,f) < ε . Proof Of Necessary Part :               Let's suppose that the bounded function f be integrable i.e          b          - b            b        ∫ f dx = ∫ f dx = ∫ f dx        -a            a             a      Let ε > 0 be a positive small number . By Darboux's Theorem ∃ a δ > 0 such that for every partition P with norm μ(P) < δ we...
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