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

  1. Mar 4, 2012 #1
    Separable equations: How do you know which variable to solve for? + extra question

    1. The problem statement, all variables and given/known data
    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?

    1. The problem statement, all variables and given/known data
    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!
     

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    Last edited: Mar 4, 2012
  2. jcsd
  3. Mar 4, 2012 #2

    LCKurtz

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    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.
     
    Last edited: Mar 4, 2012
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