MATHS2

Example 13.4 Solve the Lagrange equation

      y = xp² + 2xp(dp/dx) + 2p(dp/dx)

or

      p(1-p) = 2p(x+1)(dp/dx)

hence, provided  not(p = 0) and  not(p = 1), we see that

(dx/dp) - (2/(1-p))x = 2/(1-p)

This has as its integrating factor the expression (1-p)²  and as its solution
the expression

            p(2-p)                 C           
    x = -------- + ----------
            (1-p)²            (1-p)²

Combining this result with the original differential equation shows that

                          p²
    y = (1+C) -------
                      (1-p)²