These types of functions are important in Ordinary differential equations.

**Homogeneous
Functions and Euler’s Theorem**

**Definition 8.12**

(a) Let *A* = {( *x*,
*y* ) | *a* < *x* < *b*, *c* < *y* < *d* }
⊂ ℝ^{2} , *F* : *A* → ℝ, we say that *F* is a
homogeneous function on *A* , if there
exists a constant *p* such that *F* ( *λ* *x*, *λ* *y* ) = *λ* ^{p}*F* ( *x* , *y*)
for all *λ* ∈ ℝ and sutitably
restricted *λ*, *x*, *y,* such that (*λ* *x* , *λ* *y*) ∈ *A* . This constant *p* is
called degree of *F* .

(b) Let *B* = {( *x*,
*y* , *z*) | *a* < *x* < *b* ,
*c* < *y* < *d*, *u* < *z* < *v*} ⊂ ℝ^{3} , *G* : *B* → ℝ, we say that *G* is a
homogeneous function on *B* , if there
exists a constant *p* such that *G* ( *λ* *x*, *λ* *y* , *λ* *z* ) = *λ* * ^{p}G* (

Note: Division by any variable may occur, to avoid division by zero, we
say that *λ*, *x*, *y, z* are sutitably restricted real numbers.

These
types of functions are important in Ordinary differential equations (Chapter
10). Let us consider some examples.

Consider
*F *(* x*,* y*)* *=* x*^{3}* *−* *2* y*^{3}* *+* *5*xy*^{2}* *, (*x
*,* y*)* *∈ ℝ^{2} . Then

*F *(* **λ** x *,*
**λ** y*)*
*=* *(*λ** x *)*
*^{3}* *−* *2(*λ** y*)^{3}* *+* *5(*λ** x *)(*λ** y*)^{2}* *=* **λ*^{3}* *(*x
*^{3}* *−* *2* y*^{3}* *+* *5*xy*^{2}* *)

and
hence *F* is a homogeneous function of
degree 3.

On the other
hand,

*G *(* x*,* y*)*
*=* e ^{x}*

because,* G *(*
**λ** x *,*
**λ** y*)*
*=* e *^{(}^{λ}* ^{x}*)2

for any *λ* ≠ 1 and any *p*.

**Example 8.21**

Show
that is a homogeneous function of degree 1.

**Solution**

We
compute

for all *λ* ∈ ℝ. So *F* is a
homogeneous function of degree 1.

We state
the following theorem of Leonard Euler on homogeneous functions.

** **

**Definition 8.13
(Euler)**

Suppose that *A* = {( *x*, *y* ) | *a* < *b*, *c* < *y* < *d* }⊂ ℝ^{2}, *F* : *A* → ℝ^{2} . If *F* is having
continuous partial derivatives and homogeneous on *A* , with degree *p* , then

Suppose that *B* = {( *x*, *y* , *z*)
| *a* < *x* < *b*, *c* < *y* < *d*, *u* < *z* < *v*} ⊂ ℝ^{3} , *F* : *B* → ℝ^{3}. If *F* is having continuous
partial derivatives and homogeneous on *B*
, with degree *p* , then

We omit
the proof. The above theorem is also true for any homogeneous function of *n* variables; and is useful in certain
calculations involving first order partial derivatives.

** **

**Example 8.22**

**Solution**

Note
that the function *u* is not
homogeneous. So we cannot apply Euler’s Theorem for *u*.

However,
note that is homogeneous;
because

** **

**Note:**

Solving
this problem by direct calculation will be possible; but will involve lengthy
calculations.

Tags : Mathematics , 12th Maths : UNIT 8 : Differentials and Partial Derivatives

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