How do I solve this partial differential equation: $$ a\frac{\partial I}{\partial a} + b\frac{\partial I}{\partial b} + c\frac{\partial I}{\partial c} = \frac{\pi}{2} ?$$
Is there a way to approach this partial differential equation via Lagrange's partial differential equation for 3 variables?
$$ a\frac{\partial I}{\partial a} + b\frac{\partial I}{\partial b} + c\frac{\partial I}{\partial c} = \frac{\pi}{2} $$ $$\frac{da}{a}=\frac{db}{b}=\frac{dc}{c}=\frac{dI}{\pi/2}$$ Change of variables : $$a=e^x\quad;\quad b=e^y\quad;\quad c=e^z \quad;\quad I=\frac{\pi}{2}x+J(x,y,z)$$ $$ \frac{\partial J}{\partial x} + \frac{\partial J}{\partial y} + \frac{\partial J}{\partial z} = 0 $$ I suppose that you can find (or something equivalent) : $$J(x,y,z)=F\big((x-y),(y-x),(z-x) \big)$$ $F$ is an arbitrary function until no condition is specified.
This result can be expressed on many equivalent forms of functions of $\frac{a}{b},\frac{b}{c},\frac{c}{a}$.