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         p=(-2x+3p^2)dp/dx
       =>       dx/dp + 2x/p=3p


            I.F. = e^∫2/pdp=e^logp^2 = p^2
    Multiplying both sides of equation by p^2,

we obtain


             d/dp(xp^2)=3p^3,


 Which on integration yields


           xp^2=3/4p^4+c
i.e.      x= c/p^2+3/4p^2.
 Thus, the general solution is the elimination of p between y= 2xp- p^3
  and x=c/p^2 + 3/4p^2
ABOUT SCIENTIST:

     
             Joseph-Louis Lagrange was an Italian Enlightenment Era mathematician and astronomer.

He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.