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Generalized homogeneous equations

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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.

equation left hand side x cubed times d times y divided by d times x equals right hand side y times open x squared plus y close

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:

x equals e super t comma y equals u times e super m times t

which in this case gives:

equation left hand side d times x equals right hand side e super t times d times t comma equation left hand side d times y equals right hand side e super two times t times open u super prime plus two times u close

equation left hand side d times y divided by d times x equals right hand side e super t times open u super prime plus two times u close

Hence the equation becomes:

equation left hand side e super four times t times open u super prime plus two times u close equals right hand side u times e super four times t times open one plus u close

equation left hand side u super prime plus u equals right hand side u squared

u super prime divided by u squared plus one divided by u equals one

letting z=1/y

equation left hand side d times z divided by d times x equals right hand side z minus one

z equals one plus c times x

y equals x squared divided by one plus c times x

This method can of course, be applied to higher order equations

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