PFAFFIAN Differential Equations And It's Solutions
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 --> --> X . dr = 0 ..............(2) If there exists a function μ(x,y,z) such that μ(P dx + Q dy + R dz) is an exact differential dΦ , the equation is said to be integrable and to possess an integrating factor μ(x,y,z) . In such case(1) becomes