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Homogeneous Function
1 2 2
0 1 2
Consider the function
f(x,y) = a .......... aThe degree of each term in x and y is n.
n n n n
nx a x y a x y y
Such functions are called homogenious functions of degree n.
A function f(x,y) of two independent variables x and y
is said to be homogenious of degree n if f(x,y) can be
written in the form x where can be any functionny
x
Another def.
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3 2 3
Some examples of homogenious functions
(1) : F(x,y)= sin( )
(2) : F(x,y)= 3
(3) : F(x,y)=
n yxx
x xy y
y x
y x
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Eulers Theorem on Homogeneous Function
If z = F (x,y) be a homogenious function of x,y of degree n
then x + y = nz for all x,yz z
x y
Proof: We have
z is a homogenious function of degree n.
so that z = xny
x
1
2
1 2
( ) '( )
( ) '( )
n n
n n
z y y ynx f x f
x x x x
y y
nx f yx f x x
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11, '( ) '( )
n nz y ySimilarly x f x f
y x x x
1 1
Thus ,we have
x + y = ( ) '( ) '( )
n n nz z y y y
nx f yx f yx f x y x x x
x + y = ( ) =nz
hence the result.
nz z ynx f
xx y
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2 2 22 2
2 2
COROLLARY I:
If z = ( , ) is a homo. function of x and y of degree n,
then + 2xy ( 1)
f x y
z z zx y n n zx x y y
Eular's theorem,we have
x + y = nz
By
z z
x y
2 2
2
Differentiating partially w.r.t.x, we get
+x + y =nz z z z
x x x y x
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2 2
2
Again differentiating partially w.r.t.y,we get
x + + y = n
z z z z
x y y y y
2 2 2
2 22 2
ultiplying by x and y respectivily and add
x
M
z z z z zx xy y yx x x y y y
2
x + y
z z
n n zx y
2 2 2
2 2 2
2 2( 1)
z z zx xy y n z nz n n z
x x y y
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Example : Verify Euler's theorem for the function
z = logny
xxSolution:
z is a homogenious function of x and y of degree n.
x + y = nzz z
x y
1
2
1 1
, log *
log
n n
n n
z y x yNow nx x
x x y x
ynx x
x
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Multiply by x and y and add
1 1log *
= log
= log
= n z
nn n
n n n
n
z z y xx y x nx x y
x y x yy
n x x xx
yn x
x
1* *
nnz x xand x
y y x y
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2 2 2
2 2 2
2 2 2
2 2 2
1Example : If u= 0
then show that 0
and x y zx y z
u u u
x y z
2 2 21
Solution : We have u =x y z
3 2
2 2 2
3 22 2 2
1 22
= -x
u x y z xx
x y z
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23 2 5 2
2 2 2 2 2 2 2
2
2 3 2 5 22 2 2 2 2 2 2
2
imilarly 3
3
ux y z y x y z
y
u x y z z x y zz
2 2 2
2 2 2
on adding we get
0u u u
x y z
2
3 2 5 22 2 2 2 2 2 2
23
ux y z x x y z
x
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2 21
Example : If u= sin ,
then show that tan
x y
x y
u ux y ux y
2 21
2 2
Solution : We have u = sin
z = then sin u = z
x y
x y
x yLet
x y
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By Eular's theorem ,we havez z
x y zx y
sin cos
sin cos
z uBut u u
x x x
z uand u u
y y y
2
22 2 1
where z = homogenious
1
function of degree one
y
xx yx is a
yx y x
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hence cos =sinu
or tan
z z u ux y z u x y
x y x y
u ux y ux y
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1Example : If u = cot , then show that
1sin 24
x y
x y
u ux y ux y
1Solution : We have u= cot
z = then cot u = z
x y
x y
x yLet
x y
1
2
1
where z = homogenious
1
function of degree half
y
x y xx is ax y y
x
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By Eular's theorem ,we have
= n z =
2
z z zx y
x y
2
2
cos
cos and we have
z uec x
x x
z uand ec xy y
2 2
2
1cos cos = cot
2
cot 1= sin 2
2cos 4
u ux ec x y ec x u
x y
u u ux y u
x y ec x
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3 31
2 2 22 2 2
2 2
Example : If u= tan , x y then show that
2 1 4sin sin 2
x y
x y
u u ux xy y u ux x y y
3 3
1
3 3
3
3 3 32
Solution : We have u=tan
z = then tan u = z
1where z = homogenious
1
function of degree two.
x y
x y
x yLet
x y
yx y xx is ayx y
x
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22 22 2
2 2
22 22 2
2 2
2 22 2
lso sec +2 sec tan u
sec +2 sec tan u
sec +2 sec tan u
z u uA u u
x x x
z u uu u
y y y
z u u uu ux y x y x y
2 2 22 2
2 2
Also by corollary of Eular's theorem,
2 2(2 1)
z z z
x xy y zx x y y
By Eular's theorem ,we have
= n z = 2 zz z
x y
x y
2 2but sec and secz u z u
u ux x y y
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2 2 22 2 2
2 2
22
2 2 2
sec 2
+2sec tan u 2 * 2 tan u
u u uu x xy y
x x y y
u u u uu x xy y
x x y y
22 2 22 2
2 2
2 2 2
2 2 22 2
2 2 tan u 2sin cos u
2 sin 2 2 tan u sin 2
=sin 2 1 2 tan u sin 2
u u u u ux xy y x y u
x x y y x y
u u ux xy y u ux x y y
u u
2
=sin 2 1 4sinu u
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Exercise
1
2 2
1 1
1 Find the first order partial derivatives of
(a) cot (x+y)(b) sin( )
(c)
2 Find the second order partial derivatives of
(a) tan tan tan
x y
x y
x y
x y
2 2
1
1log tan
eyx
xyb
x yc x y
d
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2 2
1
2
3 Varify that
where u is log (ysinx+xsiny)
4 If z= sin , show that
*
5 If z = f(x+ay)+g(x-ay), show that
u u
x y y x
x y
x y
z y z
x x y
zy
2
22 2za x
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3
2 2 2 2
2 2 2
2 2 2
2 2
2
2 2
6 If v= , show that
0
7 If z(x+y)= , show that
4 1
8 If z = log , show that
x
x y z
v v vx y z
x y
z z z z
x y x y
x yx y
z
x
1z
y
y
3
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33 2 2
22 2 2
2 2
2
2 2
1
9 If z = 3xy - y 2 , show that
110 If u = show that
1 2
1 + 0
11 If z = tan , then sh
y x
z z z
x y x y
xy y
u zx y
x x y y
y
x
2 2
2 2
ow that
0
z z
x
