Separable equations: How do you know which variable to solve for?

[JustinLiang]

Separable equations: How do you know which variable to solve for? + extra question

Homework Statement

I attached a sample problem with variables u and t. How do I know what the answer should be at the end? In terms of u or in terms of t or it doesn't matter?

Homework Statement

I also have another question, so to prevent spam I will also post it in this thread:

The second picture I attached is the question and I ended up with the answer (as shown in my picture):
y = ±(|2a|sinx/(√3/2))-a

For this question I am confused about the plug or minus sign conventions when you remove absolute value brackets. The answer key shows:
y = (4asinx/(√3/2))-a
It doesn't even have the plus or minus sign or the absolute value brackets for 2a.

Is my answer correct as well?

Thanks!

 

[LCKurtz]

Your second question is usually handled like this. Before you put in the initial condition you have$$
\ln|a+y| = \ln(\sin x) +C$$You don't need absolute value signs in the sine because x is restricted to $(0,\pi/2)$.$$
\ln\frac{|a+y|}{\sin x} = C$$ $$
|a+y| = (\sin x) e^C$$ $$
a+y = \pm e^c \sin x$$which is similar to your steps. At this point $C$ can be any real number so $e^C$ can be any positive number, so $\pm e^C$ can be any nonzero number. So just call it a new constant $K$ and you get$$
a+y = K\sin x$$Now put in your initial conditions.

For your first question, you are given an equation with $du/dt$ and an initial condition for u as a function of t. That suggest solving for u if you can.