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Valentin Fadeev

Facts about integrating factors

Wednesday, 14 Apr 2010, 23:40
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Edited by Valentin Fadeev, Sunday, 18 Sept 2011, 23:26

Even if you are faced with a plain separable ODE, the process of separation of variables itself implies multiplying both parts by some factor. Thus the integrating factor seems to be one of the most devious tricks of solving equations.

There is a general path to establish its existence. It can be found in many textbooks. I am interested in some particular cases here which give beautiful solutions.

First, for a homogeneous equation it is possible to find a closed formula for the integrating factor.

It can be shown that for equation

cap m of x comma y times d times x plus cap n of x comma y times d times y equals zero ,

where M and N are homogeneous functions of their arguments integrating factor has the form:

mu equals one divided by x times cap m of x comma y plus y times cap n of x comma y

Apply this to equation:

open a squared times x minus y close times d times x plus open x plus y close times d times y equals zero

equation sequence mu equals one divided by x times open a squared times x minus y close plus y times open x plus y close equals one divided by a squared times x squared plus y squared

Multiplying both parts by this expression we obtain:

a squared times x minus y divided by a squared times x squared plus y squared times d times x plus x plus y divided by a squared times x squared plus y squared times d times y equals zero

Rearranging:

x times d times y minus y times d times x divided by a squared times x squared plus y squared plus a squared times x times d times x plus y times d times y divided by a squared times x squared plus y squared equals zero

d of x divided by y divided by a squared postfix plus left parenthesis x divided by y times right parenthesis squared plus one divided by two times d times open a squared times x squared plus y squared close divided by a squared times x squared plus y squared equals zero

And the result becomes obvious.

For the next example it is useful to note the fact that if mu is an integrating factor for equation cap m times d times x plus cap n times d times y equals zero giving solution in the form cap u of x comma y equals cap c , then equation left hand side mu sub one equals right hand side mu times phi of cap u where phi is any differentiable function shall also be an integrating factor. Indeed

equation sequence mu sub one times open cap m times d times x plus cap n times d times y close equals phi of cap u times mu times open cap m times d times x plus cap n times d times y close equals phi of cap u times d times cap u

giving the differential for the function normal cap phi of cap u equals integral phi of cap u d cap u

This leads to the following practical trick of finding the factor. All terms of the equations are split in two groups for each of which it is easy to find the integrating factor. Then each factor is written in the most general form involving an arbitrary function as described above. Finally we try to find such functions that make both factors identical.

Consider the following equation:

open x squared times y squared minus one close times d times y plus two times x times y cubed times d times x equals zero

Rearranging the terms:

open x squared times y squared times d times y plus two times x times y cubed times d times x close minus d times y equals zero

For the second term now the integrating factor is trivial, it is 1. Hence the most general form will look like mu sub one equals phi of y .

For the first part it is easy to see that the factor should be one divided by x squared times y cubed giving solution x squared times y equals cap c , hence equation left hand side mu sub two equals right hand side one divided by x squared times y cubed times psi of x squared times y .

To make the two identical we want mu sub two to be independent of x. Setting psi of t equals t gives equation left hand side mu sub two equals right hand side one divided by y squared .

Applying this one we get:

open x squared times d times y plus two times x times y times d times x close minus d times y divided by y squared equals zero

x squared times y plus one divided by y equals cap c

Both methods were discovered in the classic book "A course on differential equations" by V.V. Stepanov

Tags: differential equations, integrating factor, m820
Permalink 2 comments (latest comment by Valentin Fadeev, Thursday, 15 Apr 2010, 16:02)
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