Special Types Of First_Order Equations Part_1
Special Types Of First_Order Equations Part_1 : Standard Form 1 : Equations involving only p and q ,The equations of this form are f(p,q) = 0 .......(1) Charpit's equations take the forms dx/[∂f/∂p] = dy/[∂f/∂q]=dz/[p∂f/∂p+q∂f/∂q] = dp/0 = dq/0 ∴ dp = 0 ⇒p = constant = a (say) .......(2) Substituting in equation(1) , we get f(a,q) = 0 ......(3) which gives q = φ(a) = constant Hence dz = pdx + qdy = a dx + φ(a) dy Which on integration yields z = ax + φ(a)y + c .......(4) Example _ 1 : Find the complete integral of the following equation . pq = 1 Solution : pq = 1 F(p,q) = pq - 1 ..........(1) ∴ ∂F/∂p = q , ∂F/∂q = p Now , dx/[∂F/∂p] = dy/[∂F/∂q] = dz /