Mathquery

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 ...

Clairaut's Differential Equations :           The equation y= px+ f(p) .  .    .  .   . (1) of first degree in x and y is called the ' Claitaut's equation ' after the name of Alex Claude CLAIRAUT(1713_1765).        Differentiating w.r.t x, bearing in mind that p= dy/ dx, we obtain          p= xdp/dx+ p+ f(p)dp/ dx or     [x+f(p)]dp/dx=0. Equating each factor to zero, we get     dp/ dx=0.      .....(2) and x+ f(p)=0.     .......(3) Integrating (2) we obtain p=c ( a constant). Putting this value of p into (1), we find its complete integral    y= cx+ f(c).       ............(4)    which represents geometrically, a family of straight lines.   Elimination of p between (3) and (1) leads to a singular solution.   Example :           ...