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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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Cauchy Riemann Equations For Analytic Function

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Cauchy - Riemann Equations :- Cauchy-Riemann Equation Necessary Conditions For A Function To Be Analytic : Cauchy-Riemann Equation Statement :- Cauchy-Riemann Equation     The necessary conditions for  w = f(z) = u(x,y) + i v(x,y) to be analytic (differentiable) at any point z = x + i y of its domain D is that the four patial derivatives ∂u/∂x , ∂u/∂y ,∂v/∂x ,∂v/∂y should exists and satisfy the C-R partial differential equations   .          ∂u/∂x = ∂v/∂y  and  ∂u/∂y = - ∂v/∂x Proof :- Cauchy-Riemann Equation       Let f(z) = u(x,y) + i v(x,y) be analytic at any point z of its domain , then               f'(z) = lim     f(z+δz) - f(z) / δz                         δx-->0         exists and is unique . i.e it i...
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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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