Question

If u=sin-1(x³+y³/x+y) prove that dx/dy=2tanu

Answer 1

Good Morning!!

This question can be solved by using "EULER'S THEOREM".

EULER'S THEOREM :- If a function say F(x,y) is homogeneous of n degree then ACCORDING TO EULER'S THEOREM.

x dF(x,y )/dx + y dF(x,y)/dy = n F (x,y)

GIVEN FUNCTION IS

U = Sin-¹ { (x³ + y³)/ (x + y) }

Sin u = { (x³ + y³) / (x + y) }

F(x,y) = Sin u = { (x³ + y³) / (x + y) }

BEFORE APPLYING EULER'S THEOREMFIRST WE, FIRST CHECK WETHER THIS FUNCTION IS HOMOGENEOUS OR NOT.

F(kx,ky) = Sin u = { (kx)³ + (ky)³ / (kx + ky)}

F(kx,ky) = Sin u = k² { (x³ + y³) / (x + y) }

F(kx,ky) = Sin u = k² F (x,y)

So, Given function. is homogeneous of degree two.

BY USING EULER'S THEOREM WE HAVE

x d (Sin u)/dx + y d ( Sin u )/dy = 2 Sin u

x Cos u du/dx + y Cos u du/dy = 2 Sin u

x du/dx + y du/dy = 2 Tan u

HENCE PROVED.