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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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Laplace Transform Of Periodic Functions

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Laplace Transform Of Periodic Functions : Definition :            A function f(t) is said to be periodic , with a period ' l ' , if it satisfies the functional equation                   f(t±l) = f(t)     So, if t>0 , a periodic function f(t) can be written as            f(t) = f(t + nl) , n= 0,1,2.. ......(1)      For example sin t = sin(t+2πn) , n=0,1,2.... is a periodic function with period 2π .       In case a function f(t) is periodic , the Laplace transform can be expressed as an integral over one cycle of the function instead of an integral over an infinite range.            The transform of equation(1) is                       ∞           f̅(p) = ∫ e⁻ᵖᵗ f...
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