Differential Equations Reducible To Homogeneous Form

Go back to  'Differential Equations'

Equation 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

\[\frac{{dy}}{{dx}} = f\left( {\frac{{ax + by + c}}{{dx + cy + f}}} \right) \qquad \qquad \ldots (1)\]

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

\[\begin{array}{l}x \to X + h\\y \to Y + k\end{array}\]

and select h and k so that

\[\left. \begin{array}{l}ah + bk + c = 0\\dh + ek + f = 0\end{array} \right\} \qquad \qquad \ldots (2)\]

This can always be done \(\left( {{\rm{if }}\frac{a}{b} \ne \frac{d}{e}} \right).\) The RHS of the DE in (1) now reduces to

\[\begin{align}&\quad\;\;\; f\left( {\frac{{a(X + h) + b(Y + k) + c}}{{d(X + h) + e(Y + k) + f}}} \right)\\& = f\left( {\frac{{aX + bY + (ah + bk + c)}}{{dX + eY + (dh + ek + f)}}} \right)\\&= f\left( {\frac{{aX + bY}}{{dX + eY}}} \right) \qquad \qquad \qquad (\rm{using\,}(2))\end{align}\]

This expression is clearly homogeneous! The LHS of (1) is \(\frac{{dy}}{{dx}}\) which equals \(\begin{align}\frac{{dy}}{{dY}} \cdot \frac{{dY}}{{dX}} \cdot \frac{{dX}}{{dx}}\end{align}.\) Since \(\begin{align}\frac{{dy}}{{dY}} \cdot\frac{{dx}}{{dX}} = 1\end{align},\) the LHS \(\frac{{dy}}{{dx}}\) equals \(\frac{{dY}}{{dX}}.\) Thus, our equation becomes

\[\frac{{dY}}{{dX}} = f\left( {\frac{{aX + bY}}{{dX + eY}}} \right) \qquad\qquad \dots (3)\]

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.

Download SOLVED Practice Questions of Differential Equations Reducible To Homogeneous Form for FREE
Differential Equations
grade 11 | Questions Set 1
Differential Equations
grade 11 | Answers Set 1
Differential Equations
grade 11 | Questions Set 2
Differential Equations
grade 11 | Answers Set 2
Download SOLVED Practice Questions of Differential Equations Reducible To Homogeneous Form for FREE
Differential Equations
grade 11 | Questions Set 1
Differential Equations
grade 11 | Answers Set 1
Differential Equations
grade 11 | Questions Set 2
Differential Equations
grade 11 | Answers Set 2
Learn from the best math teachers and top your exams

  • Live one on one classroom and doubt clearing
  • Practice worksheets in and after class for conceptual clarity
  • Personalized curriculum to keep up with school