What does in:variable1;in:variable2; mean?

Question

A question in my ODE textbook is as follows.

Determine whether the given first-order differential equation is linear in the indicated dependent variable.

$u dv+(v+uv-e^u)du = 0;$ in v; in u;

I feel as though I would be able to solve the question if I knew what the in v; in u; means.

In this question, what does in v; in u; mean? Is this a standard notation, and if so, what is it called?

Answer 1

This is really a question about English writing. The question could be rewritten as

Determine whether the first-order differential equation $$u dv+(v+uv-e^u)du = 0$$ is linear in $u$. Determine also whether it is linear in $v$.

Answer 2

$u~dv+(v+uv-e^u)~du=0$

$u\dfrac{dv}{du}+(u+1)v-e^u=0$

$\dfrac{dv}{du}+\dfrac{u+1}{u}v=\dfrac{e^u}{u}$

$\therefore$ This is a first-order ODE with the dependent variable $v$ .

Answer 3

The notation in question looks to me like a shorthand way of asking these two questions:

1. Is $u dv+(v+uv-e^u)du = 0$ linear in $u$?

2. Is $u dv+(v+uv-e^u)du = 0$ linear in $v$?

Here I'm reading the word "in" as just a word in (mathematical) English, not a mathematical notation.