Concept: Where does the (-) come from

[Rachel Mandel]

Still confused about where the negative sign comes in the Constant Coefficients and Variable Coefficients section. For example, how I derive the example in the variable section would be...

U_t + tU_x     = 0
dx + tut      = 0
dx              = -tut
x                = -¹/₂t²
x + ¹/₂t²  = 0

[Victor Ivrii]

Do you mean $u_t + tu_x=0$? Then $\frac{dt}{1} = \frac{dx}{t}=\frac{du}{0}$. The first equation implies $$dx=tdt \implies x=\frac{1}{2}t^2+C\implies x-\frac{1}{2}t^2=C.$$

[Rachel Mandel]

Yes, I do mean that equation. When I try it myself however, (lines 2 to 3 of my initial post) the math comes out that dx = -tdt.

[Wanying Zhang]

I guess that since $u$ is of $x$ and $x$ is of $t$, they are not independent and you need to use $\partial$ in your steps. Also, there should be chain rule in this case, so you cannot simply take integrals as in your second equation.

[Victor Ivrii]

Yes, I do mean that equation. When I try it myself however, (lines 2 to 3 of my initial post) the math comes out that dx = -tdt.

You need to read Section 2.1 of textbook to understand that equation $au_t+bu_x=f$ requires equation of integral curves $\frac{dt}{a}=\frac{dx}{b}=\frac{du}{f}$. What is on your post are incomprehensibly  written senseless manipulations.