Question

If u=xyf(y/x) show that x(du/dx)+y(du/dy)=2u

Answer 1

Answer:

x($\frac{du}{dx}$)+y($\frac{du}{dy}$)=2u...... Proved.

Step-by-step explanation:

We have

u= xyf($\frac{y}{x}$) .....(1)

And We have to prove that

x($\frac{du}{dx}$)+y($\frac{du}{dy}$)=2u  .....(2)

Now, partially differentiating equation (1) with respect to x, we get

$\frac{du}{dx}$=yf($\frac{y}{x}$)+yxf'($\frac{y}{x}$)($\frac{-y}{x^{2} }$)

=yf($\frac{y}{x}$)-$\frac{y^{2} }{x}$f'($\frac{y}{x}$)..... (3)

Now, partially differentiating equation (1) with respect to y, we get

$\frac{du}{dy}$=xf($\frac{y}{x}$)+xyf'($\frac{y}{x}$)×($\frac{1}{x}$)

=xf($\frac{y}{x}$)+yf'($\frac{y}{x}$)..... (4)

So, LHS of equation (2)

=x($\frac{du}{dx}$)+y($\frac{du}{dy}$)

=xyf($\frac{y}{x}$)-y²f'($\frac{y}{x}$)+xyf($\frac{y}{x}$)+y²f'($\frac{y}{x}$)

=2xyf($\frac{y}{x}$)

=2u

=RHS of equation (2)

Hence, proved

Answer 2

Answer:

Step-by-step explanation: