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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.