1. If u=cos [ xy+ yz+ zxx2+ y2+z2 ] , then show that ,

x ∂u /∂x+ y ∂u/∂ y+¿ z∂ u/∂ z=0¿

2. Verify Euler’s Theorem for u=( x12+ y

12 )(xn+ yn)

3. If u=xφ ( yx )+ψ ( yx ) , then show that,

(a)x ∂u /∂x+ y ∂u/∂ y=¿ x φ( yx)¿

(b)x2∂2u/∂ x2+2xy ∂2u /∂x ∂ y+ y2∂2u/∂ y2=0

4. If f ( x , y )= 1x2 + 1

xy+( log x−log y)/ x2+ y2 , then show that,

x ∂ f /∂ x+ y∂ f /∂ y+¿2 f (x , y )=0¿

5. If u= x2 y2 z2

x2+ y2+ z2+cos [ xy+ yz

x2+ y2+z2 ] , then prove that,

x ∂u /∂x+ y ∂u/∂ y+¿ z∂ u/∂ z= 4 x2 y2 z2

x2+ y2+z2 ¿

6. If ¿ xn f 1( yx )+ y−n f 2( xy ) , then prove that

x2 ∂2 z∂ x2 +2 xy ∂2 z

∂ x∂ y+ y2 ∂2 z

∂ y2 +x ∂ z∂ x

+ y ∂ z∂ y

=n2 z

7. If f(x,y) and φ (x , y ) are homogeneous functions of x , of degree m and n respectively and u=f ( x , y )+φ (x , y ), then show that,

f ( x , y )= 1m(m−n) ( x2 ∂2u

∂x2 +2xy ∂2u∂ x ∂ y

+ y2 ∂2u∂ y2 )− (n−1)

m(m−n) (x ∂u∂ x + y ∂u∂ y )

8. If u=tan−1( y2

x ), then prove that,

x2∂2u∂ x2 + 2 xy∂2u

∂ x∂ y+ y

2∂2u∂ y2 =−sin 2u sin 2u

9.If z=xyf ( yx ) , then show that , x ∂ z∂ x+ y∂ z∂ y

=2 z

10. If V=log (sin ( П (2 x2+ y2+xz )1 /2

2 (x2+xy+2 yz+z2 )1 /3 )), then prove that when x=0, y=1, z=2, that,

x ∂V∂x

+ y ∂V∂ y

+z ∂V∂z

= П12