Q-1) Find dydx when (i) x y+ yx=cand (ii) (Cos x)y=(Sin y)x.

Q-2) Prove that ∂2 z∂x2 +

∂2 z∂ y2 =

∂2 z∂u2 +

∂2 z∂v2 where x=ucosα−¿vsinα and y=usinα+¿ucosα

(Hint:- z is a function of u and v and α is a constant)

Q-3) If f(x,y) = 0 and ϕ(x,y) = 0 touch each other , show that at point of contact ,

∂ f∂x∂ϕ∂ y

− ∂ f∂ y

∂ϕ∂x

=0

Q-4) If u=f(r) ,r2= x2+y2+z2 , then prove that ,

∂2 v∂x2 +

∂2 v∂ y2 +

∂2v∂ z2 =f

' ' (r )+ 2rf '(r)

Q-5) If z=xyf( yx ) and z is a constant, then prove that ,

f ' ( yx )f ( yx )

=

x ( y+x ( dydx ))y ( y−x ( dydx ))

Q-6) If z = z(u,v) , u= x2-2xy-y2 , v = y , then prove that ,

( x+ y ) ∂ z∂ x

+( x− y ) ∂ z∂ y

=0is equivalent to ∂ z∂v=0.

Q-7) By changing independent variables x and y to u and v by u=x-ay and v=x+ay , prove that

a2 ∂2 z∂x2 −

∂2 z∂ y2 transforms into 4 a2 ∂2 z

∂u∂v.

Q-8) If u = x2-y2 , v = 2xy , f(x,y) =ϕ(u,v) , then prove that ∂2 f∂ x2 +

∂2 f∂ y2 =4 (x2+ y2 )( ∂2ϕ

∂u2 +∂2ϕ∂v2 ) .