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Math Help - Simple Separation of Variable Problem

  1. #1
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    Simple Separation of Variable Problem

    I'm just starting a course in DiffEQ and we have a simple separation of variables problems.

    The problem is to change the dy/dt equation into a simple y(t) equation (example dy/dt=ty becomes y=k*e^(t^2)) where k is any real number).

    The problem I am stuck on is dy/dt=t/(y+y*t^2). Here is my work.
    dy/dt=t/(y(1+t^2))
    y*dy=[t/(1+t^2)]*dt
    Integrate both sides
    1/2y^2=1/2[ln|t^2+1|]+C where C is a constant
    y^2=ln|t^2+1| + C where C is a constant
    y = sqrt (ln|t^2+1| + C) where C is a constant and the sqrt is either positive or negative.

    The book gives an answer of y(t)=sqrt(ln|k(t^2+1)|) where k is any real number and the sqrt is either positive or negative.

    I understand most of the steps, but why do I get ln|t^2+1| + C inside the square root while the book gets ln|k(t^2+1)| inside the square root?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by David_is_a_LOSTaway View Post
    I'm just starting a course in DiffEQ and we have a simple separation of variables problems.

    The problem is to change the dy/dt equation into a simple y(t) equation (example dy/dt=ty becomes y=k*e^(t^2)) where k is any real number).

    The problem I am stuck on is dy/dt=t/(y+y*t^2). Here is my work.
    dy/dt=t/(y(1+t^2))
    y*dy=[t/(1+t^2)]*dt
    Integrate both sides
    1/2y^2=1/2[ln|t^2+1|]+C where C is a constant
    y^2=ln|t^2+1| + C where C is a constant
    y = sqrt (ln|t^2+1| + C) where C is a constant and the sqrt is either positive or negative.

    The book gives an answer of y(t)=sqrt(ln|k(t^2+1)|) where k is any real number and the sqrt is either positive or negative.

    I understand most of the steps, but why do I get ln|t^2+1| + C inside the square root while the book gets ln|k(t^2+1)| inside the square root?
    Define C=\ln K.
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