Toronto Math Forum
APM3462019 => APM346Lectures & Home Assignments => Home Assignment 2 => Topic started by: Rachel Mandel on January 22, 2019, 04:29:32 PM

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}/_{2}t^{2}
x + ^{1}/_{2}t^{2} = 0

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

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.

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.

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.