Question

if u=y/z+z/x, find x ∂ u /∂x +y ∂u/∂y+z ∂u/∂z​

Answer 1

Answer:

We have:

u

=

y

x

+

z

x

+

x

y

and we seek to validate that

f

satisfies the Partial differential Equation:

x

∂

u

∂

x

+

y

∂

u

∂

y

+

z

∂

u

∂

z

(In other words we are validating that a solution to the given PDE is

u

). We compute the partial derivative (by differentiating wrt to specified variable and treating all other variables as constants), and applying the chain rule:

u

x

=

∂

u

∂

x

=

−

y

x

2

−

z

x

2

+

1

y

u

y

=

∂

u

∂

y

=

1

x

−

x

y

2

u

z

=

∂

u

∂

z

=

1

x

Next we compute the LHS of the desired expression:

L

H

S

=

x

∂

u

∂

x

+

y

∂

u

∂

y

+

z

∂

u

∂

z

=

x

(

y

(

1

−

2

x

y

+

y

2

)

2

)

−

y

(

x

−

y

(

1

−

2

x

y

+

y

2

)

2

)

=

x

(

−

y

x

2

−

z

x

2

+

1

y

)

+

y

(

1

x

−

x

y

2

)

+

z

(

1

x

)

=

−

y

x

−

z

x

+

x

y

+

y

x

−

x

y

+

z

x

Noting that all terms cancel, we then have the desired result:

L

H

S

=

0

=

R

H

S

QED