Question

Find the solution of the following symmetrical simultaneous differential equation dx/1 = -dy/1 = dz/1

Answer 1

$dx/1 = -dy/1 = dz/1$

$dx = -dy = dz$

$d(x+y)=d(z+y)=0$

Therefore, the functions $x+y$ and $z+y$ are constant along any integral curve and form a complete system of integral invariants. Therefore, the general solution can be written as

$\Phi(x+y,z+y)=0$, where $\Phi$ is an arbitrary smooth function of two variables.