CHAPTER 9 - Equations Reducible to Homogeneous Form
Many a times, the DE specified may not be homogeneous but some suitable manipulation might reduce it to a homogeneous form. Generally, such equations involve a function of a rational expression whose numerator and denominator are linear functions of the variable, i.e., of the form
Note that the presence of the constant and causes this DE to be non-homogeneous. To make it homogeneous, we use the substitutions
and select and so that
This can always be done . The RHS of the DE in now reduces to
This expression is clearly homogeneous ! The LHS of is which equals . Since the LHS equals . Thus, our equation becomes
We have thus succeeded in transforming the non-homogeneous DE in to the homogeneous DE in . This can now be solved as described earlier. Let us apply this technique in some examples.