I haven't studied separation of variables in nonlinear equations yet. I was wondering if anyone could offer any hints to the question below.
The Eikonal equation for light rays in geometric optics is $(∂u/∂x)^2+(∂u/∂t)^2=1$ Find the general solution by separating variables with the sum $u(x,t)=X(x)+T(t)$.
I have looked online but I don't understand so was hoping someone would be able to explain it to me in layman terms.
If you try substituting the given form $u(x,t) = X(x) + T(t)$ into the differential equation, you get $X'(x)^2 + T'(t)^2 = 1$, or $X'(x)^2 = 1 - T'(t)^2$. This has to hold for all $x$ and $t$. But the left side doesn't depend on $t$, while the right side doesn't depend on $x$, so both sides must be constant: $X'(x)^2 = c = 1 - T'(t)^2$. Thus $X'(x) = \pm \sqrt{c}$ and $T'(t) = \pm \sqrt{1-c}$, where $0 \le c \le 1$. And then integrating $X'$ and $T'$ to get $X$ and $T$, we have the solution $$ u(x,t) = a \pm \sqrt{c}\; x \pm \sqrt{1-c}\; t $$ where $a$ is another arbitrary constant.