Solving for the Non-homogeneous Equation

Date: 10/21/2001 at 19:07:05
From: S. Parker
Subject: Non-homogeneous DE

I have the equation:

x^2(d^2y/dx^2) - 3x(dy/dx) = 3x^3

I attempted to solve and have the general solution for the homogeneous 
equation. But I am stuck after that - how do I continue solving for 
the non-homogeneous equation?

Date: 10/24/2001 at 19:48:43
From: Doctor Anthony
Subject: Re: Non-homogeneous DE

This is an example of Euler's linear equation.

Put x = e^t

Then   x(dy/dx) = dy/dt

       x^2(d^2y/dx^2) = d^2(y)/dt^2 - dy/dt

Substitute into the differential equation

    d^2(y)/dt^2 - dy/dt - 3(dy/dt) = 3e^(3t)

    d^2(y)/dt^2 - 4(dy/dt) = 3e^(3t)

   If dy/dt = p   d^2/dt^2 = dp/dt we get

        dp/dt - 4p = 3e^(3t)

          (D-4)p = 3.e^(3t)

The CF is  p = A.e^(4t)

The PI is  p = 3.e^(3t)/(D-4)

             = 3.e^(3t)/(3-4)

             = -3.e^(3t)

So the general solution is

         p = A.e^(4t) - 3.e^(3t)

       dy/dt = A.e^(4t) - 3.e^(3t)

         y = (A/4)e^(4t) - (1/9)e^(3t) + E

         y = C.x^4 - (1/9)x^3 + E

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