Second Order Differential Equation

Date: 11/04/2001 at 02:52:04
From: Roger
Subject: 2nd order differential equations

By letting p = dy/dx,

solve

y(d2y/dx2) = 2 (dy/dx)2

This is what I have done:

d2y/dx2 = dp/dx
y (dp/dx) = 2p2...
...
...

Date: 11/04/2001 at 08:08:51
From: Doctor Anthony
Subject: Re: 2nd order differential equations

    y.(d^2y/dx^2) = 2(dy/dx)^2

Since x is absent explicitly,

put  dy/dx = p   and  d^2y/dx^2 = p.(dp/dy)

[We can prove this last expression from

          p = dy/dx

      dp/dy = d^2(y)/dx^2.(dx/dy)

            = (1/p).d^2(y)/dx^2

  p.(dp/dy) = d^2(y)/dx^2 ]


Our original equation can now be written

 y[p.dp/dy] = 2p^2

       dp/p = 2dy/y

      ln(p) = 2.ln(y) + ln(A)

      ln(p) = ln[Ay^2]

          p = A.y^2

      dy/dx = A.y^2

     dy/y^2 = A.dx

       -1/y = A.x + B   

          y = -1/(Ax+B)


- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/