This is quite a minor trick and like many things listed here may seem quite trivial. However, this is one of those few occasions when I had the tool in mind, before I actually got the example touse it on. Consider:
which does not really require a great effort to solve. But forget all the standard ways for a moment and add to both parts:
Hope this can be stretched to use in more complicated cases
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
,
where M and N are homogeneous functions of their arguments integrating factor has the form:
Apply this to equation:
Multiplying both parts by this expression we obtain:
Rearranging:
And the result becomes obvious.
For the next example it is useful to note the fact that if is an integrating factor for equation giving solution in the form , then where is any differentiable function shall also be an integrating factor. Indeed
giving the differential for the function
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:
Rearranging the terms:
For the second term now the integrating factor is trivial, it is 1. Hence the most general form will look like .
For the first part it is easy to see that the factor should be giving solution , hence .
To make the two identical we want to be independent of x. Setting gives .
Applying this one we get:
Both methods were discovered in the classic book "A course on differential equations" by V.V. Stepanov
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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