Edited by Valentin Fadeev, Sunday, 16 Jan 2011, 23:20
One of the rarely used methods of solving ODEs applies to the so-called generalized homogeneous equations. The word "generalized" means that the terms are not homogeneous in the classic sense, if all variables are assigned the same dimension. But they may be made homogeneous in a wider sense by choosing the appropriate dimension for the dependent variable. Here is one example.
If we assign dimension 1 to x and dx and dimension m to y and dy, then the left side has dimension 3+m-1=m+2 on the right side we have m+2 and 2m. To balance things let m+2=2m, hence m=2 and we get a "generalized homogeneous equation" of the 4th order. The trick is to let:
which in this case gives:
Hence the equation becomes:
letting z=1/y
This method can of course, be applied to higher order equations
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