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Tuesday, 5 August 2014

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
\dfrac{{dy}}{{dx}} = f\left( {\dfrac{{ax + by + c}}{{dx + cy + f}}} \right)\ldots (1)
Note that the presence of the constant c and f causes this DE to be non-homogeneous. To make it homogeneous, we use the substitutions
x \to X + h
y \to Y + k
and select h and k so that
\left. \begin{array}{l}  ah + bk + c = 0\\  dh + ek + f = 0  \end{array} \right\}\ldots (2)
This can always be done \left( {{\rm{if }}\dfrac{a}{b} \ne \dfrac{d}{e}} \right). The RHS of the DE in (1) now reduces to
f\left( {\dfrac{{a(X + h) + b(Y + k) + c}}{{d(X + h) + e(Y + k) + f}}} \right)
 = f\left( {\dfrac{{aX + bY + (ah + bk + c)}}{{dX + eY + (dh + ek + f)}}} \right)
 = f\left( {\dfrac{{aX + bY}}{{dX + eY}}} \right)Using (2)
This expression is clearly homogeneous ! The LHS of (1) is \dfrac{{dy}}{{dx}} which equals \dfrac{{dy}}{{dY}} \cdot \dfrac{{dY}}{{dX}} \cdot \dfrac{{dX}}{{dx}}. Since the LHS \dfrac{{dy}}{{dx}} equals \dfrac{{dY}}{{dX}}. Thus, our equation becomes
\dfrac{{dY}}{{dX}} = f\left( {\dfrac{{aX + bY}}{{dX + eY}}} \right)\ldots (3)
We have thus succeeded in transforming the non-homogeneous DE in (1) to the homogeneous DE in (3). This can now be solved as described earlier. Let us apply this technique in some examples.
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