Question

a) Verify Euler's theore
for f(x,y) = x^3 +2y^3- x^2y
​

Answer 1

Answer:

Euler's theorem

f(x,y)=

x

2

+y

2

1

f(tx,ty)=

t

2

x

2

+t

2

y

2

1

=

t

1

.f(x,y)=t

−1

f(x,y)

∴ f is a homogeneous function of degree −1 and by Euler's theorem

x

∂x

∂f

+y

∂y

∂f

=−f

Verification:

∂x

∂f

=

2

−1

.

(x

2

+y

2

)

3/2

2x

=

(x

2

+y

2

)

3/2

−x

Similarly

∂y

∂f

=

(x

2

+y

2

)

3/2

−y

x

∂x

∂f

+y

∂y

∂f

=−(

(x

2

+y

2

)

3/2

x

2

+y

2

)

x

2

+y

2

−1

=−f