Special Types Of First_Order Equations Part_3
Special Types Of First_Order Equations Part_3 : Standard Form _ 3 : Separable equations i.e. equations of the type f₁(x,p) = f₂(y,q) ............ (1) The Charpit's auxiliary equation for equation(1) are dx/[∂f₁/∂p] = dy/-[∂f₂/∂q] = dz/[p∂f₁/∂p - q∂f₂/∂q] = dp/-[∂f₁/∂x] = dq/-[∂f₂/∂y] ∴ dx/[∂f₁/∂p] = dp/-[∂f₁/∂x] ⇒ ∂f₁/∂x dx +∂f₁/∂p dp = 0 ⇒df₁(x,p) = 0 Integrating , f₁(x,p) = a (say) ......... (2) Hence , equation(1) gives f₂(y,q) = a .......... (3) Solving equations (2) and (3) for p and q , we obtain p = g₁(a,x) and q = g₂(a,y) Now dz = pdx + qdy gives dz = g₁(a,x) dx + g₂(a,x) dy Integrating , z = ∫g₁(a,x) dx +∫g₂(a,y) dy +c ............... (4) Where c is an arbitrary constant . Examples Related To