Special Types Of First _ Order Equations (Part _2)
Special Types Of First Order Differential Equations (Part_2) : Standard Form _2 : Equations not involving the independent variables i.e. equation of the form f(z,p,q) = 0 ...........(1) The auxiliary equations for equation(1) are dx/[∂f/∂p] = dy/[∂f/∂q]= dz/[p∂f/∂p+q∂f/∂q] = dp/-p∂f/∂z = dq/-q∂f/∂z ∴ dp/p = dq/q Integrating q = ap ............(2) Where a is a constant . Substituting in equation(1) , we have f(z,p,ap) = 0 Now dz = pdx + qdy ...............(3) = pdx + apdy = pd(x+ay) Let us take X = x + ay So that dz/dX = p and adz/dX = q . Hence the given equation becomes f(z,dz/dX,adz/dX) = 0 ...................(4) The equation (4) is an ordinary differential equation of first order and can easily be solved . The co