The Lagrange Differential Equation With Scientist Name
Hello my friends, I want discuss something about the famous equation called LAGRANGE EQUATION. This equation is used to solve differential equations of higher degree like CLAIRAUT EQUATION. Now , The equation y= xg(p)+ f(p) is associated with the name of JOSEPH LOUIS LAGRANGE (1736_1813). This is a generalised form of CLAIRAUT's EQUATION . If we put g(p) =p, it is in CLAIRAUT's form. Differentiating w.r.t x and putting dy/ dy/dx = y'= p, we have p= g(p) + xg'(p) dp/dx+f'(p) dp/dx or [p-g(p)] dx/dp= xg'(p)+ f'(p) or dx/dp= g'(p)x/p-g(p) + f'(p)/p-g(p)..(1) The equation (1) is a linear differential equation in x and dx/dp and it is integrable using different types of methods. Example: Solve y= 2xy- p^3 Solution. Differentiating, we get p= 2p+ 2xdp/dx - 3p^2 dp/dx or