Vector Calculus | Part IA, 2001

For a real function f(x,y)f(x, y) with x=x(t)x=x(t) and y=y(t)y=y(t) state the chain rule for the derivative ddtf(x(t),y(t))\frac{d}{d t} f(x(t), y(t)).

By changing variables to uu and vv, where u=α(x)yu=\alpha(x) y and v=y/xv=y / x with a suitable function α(x)\alpha(x) to be determined, find the general solution of the equation

xfx2yfy=6fx \frac{\partial f}{\partial x}-2 y \frac{\partial f}{\partial y}=6 f

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