Solution of First Order and First Degree Differential Equations: Homogeneous Form or Homogeneous Differential Equation

**Homogeneous Form
or Homogeneous Differential Equation**

A function *f* ( *x*, *y*)
is said to be a **homogeneous** function of degree *n* in the variables *x* and *y *if, *f *(*tx
*,* ty *)* *=* t ^{n} f *(

For
instance,

(i) *f *(*x*,* y*)*
*=* *6*x*^{2}* *+* *2*xy
*+* *4* y*^{2}* *is a homogeneous function in* x *and* y*, of degree two.

(ii) But
*f* (*x*, *y*) =
*x*^{3} +
(
sin *x*)*e ^{y}*
is not a homogeneous function.

If* f *(*
x*,* y*)* *is a* ***homogeneous*** *function of degree zero, then there
exists a function* g *such that *f *(*
x*,* y*) is always expressed in the
form* g*(*y/x*)* *or* g*(*x/y*).

An ordinary differential equation is said to be in **homogeneous form**, if the differential equation is written as *dy/dx* =* g*(*y*/*x*)* *.

The word
“homogeneous” used in Definition 10.7 is different from in Definition 10.12.

(i) The
differential equation *M* ( *x*, *y*)*dx* + *N*(
*x*, *y*)*dy* =
0 [in differential form] is said to be **homogeneous **if** ***M*** **and** ***N*** **are** homogeneous
functions of the same degree**.

(ii) The
above equation is also written as *dy/dx*
= *f* ( *x*, *y*) [in derivative form] where

*f *(* x*,* y*)*
*= −*M *(* x*,* y*) /*
N *(* x*,* y*)* *is clearly
homogeneous of degree* *0* *.

For
instance

(1) consider
the differential equation (*x*^{2}* *−* *3*y*^{2})*dx *+* *2*xy dy *=* *0* *. The given
equation is rewritten as .
Thus, the given equation is expressed as Hence, (*x*^{2}
- 3*y*^{2})*dx* + 2*xydy* = 0 is a
homogeneous is a homogeneous is a homogeneous
differential equation.

(2) However, the differential equation is not homogeneous. (verify!)

To find
the solution of a homogeneous differential equation *dy/dx* =* g*(*y/x*), consider the substitution *v = y/x*. Then,* y *= *xv* and* dy/dx* =* v *+* x dv/dx* .Thus, the
given differential equation becomes* x dv/dx*
=* f *(*v*)* *−* v *which is solved using variables
separable method. This leads to the following result.

** **

If *M* ( *x*, *y*
) *dx* + *N*( *x* , *y*)*dy* = 0 is a homogeneous equation, then the change of variable *y* = *vx*, transforms into a
separable equation in the variables *v*
and *x* .

Tags : Solution of First Order and First Degree Differential Equations | Mathematics , 12th Maths : UNIT 10 : Ordinary Differential Equations

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12th Maths : UNIT 10 : Ordinary Differential Equations : Homogeneous Form or Homogeneous Differential Equation | Solution of First Order and First Degree Differential Equations | Mathematics

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