Solve the differential equation:$\frac{\,dx}{mz-ny}=\frac{\,dy}{nx-lz}=\frac{\,dz}{ly-mx}$

Question

QUESTION: Solve the differential equation: $$\frac{\,dx}{mz-ny}=\frac{\,dy}{nx-lz}=\frac{\,dz}{ly-mx}$$

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MY ATTEMPT: I tried out to proceed by using $$\frac{\,dx}{mz-ny}=\frac{\,dy}{nx-lz}=\frac{\,dz}{ly-mx}=k(\text{say, a constant})$$

Then I got $$dx=k(mz-ny)$$ $$dy=k(nx-lz)$$ $$dz=k(ly-mx)$$

And then I could by elimination of variables, create diffential equations with $2$ variables, like $$l\,dx+n\,dz=k(lmz-nmx)$$ Or $$l\,dx+knmx=klmz-n\,dz$$

But then I could not proceed any further since I have no idea on how to integrate expressions like $l\,dx+knmx$ where $x$ and $\,dx$ are separated by a $+$ sign.

Can anyone help.

Answer 1

Following my comment, after substituting $nz=c-lx-my$ into $\frac{\text{d}x}{mz-ny}=\frac{\text{d}y}{nx-lz}$, you will obtain $$\left(l^2+n^2\right)x^2+2lmxy+\left(m^2+n^2\right)y^2-2clx-2cmy+k=0$$ for some constant $k$. The $(x,y)$-plot looks like an ellipse. (By symmetry, the $(y,z)$-plot and the $(z,x)$-plot are also elliptical.) In fact, the $(x,y,z)$-plot will be an ellipse on the plane $lx+my+nz=c$, and the equation above can be also written as $$l^2x^2+m^2y^2+n^2z^2=r^2$$ where $r^2:=c^2-k$. So, we have an intersection between a plane $lx+my+nz=c$ and an ellipsoid $l^2x^2+m^2y^2+n^2z^2=r^2$ in the $(x,y,z)$-space.

Answer 2

What if you consider $\frac {dx}{mz-ny}=\frac{dy}{nx-lz}$ , $\frac{dy}{nx-lz}=\frac{dz}{ly-mx}$ and $\frac{dx}{mz-ny}=\frac{dz}{ly-mx}$separately. Yielding $\int{(nx-lz)dx}=\int{(mx-ny)dy}$ , $\int{(ly-mx)dy}=\int{(nx-lz)dz}$, and $\int{(ly-mx)dx}=\int{(mz-ny)dz}$. Giving Three equations $\frac{nx^2}{2}=-\frac{ny^2}{2}$, $\frac{ly^2}{2}=-\frac{lz^2}{2}$, and $\frac{nx^2}{2}=-\frac{lz^2}{2}$.

Yielding $x^2=-y^2$, $y^2=-z^2$, and $x^2=-z^2$.

or

$x^2=-y^2=-z^2$.