$dh/dt \pm \sqrt{h} du/dt = 0 $ $\rightarrow $ $d/dt(u \pm 2 \sqrt{h}) = 0 $

Question

$dh/dt \pm \sqrt{h} du/dt = 0 $ $\rightarrow $ $d/dt(u \pm 2 \sqrt{h}) = 0 $

i cant seem to show the implication. $u=u(x,t)$ and $h=h(x,t)$

can someone please help me

Answer 1

Maybe it's ap artial derivative. Anyway we can take $x$ as a parameter.

$\dfrac{dh}{dt} \pm \sqrt{h}\dfrac{du}{dt} = 0$

$\dfrac{1}{\sqrt{h}}\dfrac{dh}{dt} \pm \dfrac{du}{dt}= 0$

$\dfrac{d}{dt}(2\sqrt{h})\pm \dfrac{du}{dt}=0$ or $\dfrac{d}{dt}(2\sqrt{h}\pm u)=0$

Answer 2

$$\frac {d}{dt}(u\pm 2\sqrt h)=\frac {du}{dt}\pm 2\frac {d}{dt}\sqrt h$$

$$=\frac {du}{dt}\pm h^{-1/2}\frac {dh}{dt}$$

$$=h^{-1/2}[\sqrt h \frac {du}{dt}\pm\frac{dh}{dt}]=h^{-1/2}(0)=0$$