If sin \(u = \frac{x^3 + y^3}{\sqrt{x} +\sqrt{y}}\), then xu_x + yu_y will be equal to:

Question

Question

This question was previously asked in

AAI ATC Junior Executive 2018 Official Paper (Shift 3)

Answer 1

Explanation:

Given:

\(\sin u = \frac{{{x^3} + {y^3}}}{{\sqrt x + \sqrt y }}\)

\(x{u_x} + y{u_y} = ?\)

Now,

u = f(x, y) but it is not homogenous but f(u) is homogenous of degree n

∴ \(x\frac{{dy}}{{dx}} + y\frac{{du}}{{dy}} = n\frac{{f\left( u \right)}}{{f'\left( u \right)}}\)

f(nu) = \(\frac{{{(nx)^3} + {(ny)^3}}}{{\sqrt nx + \sqrt ny }} = n^{5/2}.\frac{{{x^3} + {y^3}}}{{\sqrt x + \sqrt y }}\)

⇒ f(nu) = n^5/2 f(u);

Therefore, homogeneous of degree 5/2;

∴ \(x\frac{{du}}{{dx}} + y\frac{{du}}{{dy}} = \frac{5}{2} \times \frac{{\sin u}}{{\cos u}}\)

\(x{u_x} + y{u_y} = \frac{5}{2}\tan u\)