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Math Help - Explanation of Solution Help

  1. #1
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    Explanation of Solution Help

    My online class provided this solution but i have questions about some of the steps. My online professor has been no help. Any input is appreciated.

    Question Solve the initial value problem. \frac{dy}{dx}=\frac{y-1}{x+3}

    Step 1 Separate and integrate \int\frac{dy}{y-1}=\int\frac{1}{x+3}dx

    Step 2 Integration result ln(y-1)=ln(x+3)+lnC So where did the lnC come from. I know where the C came from but why are they multiplying it by ln

    Step 3 Raise all to the e e^{ln(y-1)}=e^{ln(x+3)}+e^{lnC}

    Step 4 Result is (y-1)=C(x+3) Why are they multiplying the right side by C shouldnt it be like this (y-1)=(x+3)+C

    Step 5 (y-1)=-C(x+3) why it the right side now negative?

    Step 6 y=1-C(x+3) Solution given to us by the professor.

    I cant figure out how they arrived to some of those steps maybe someone here can. Thanks.
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  2. #2
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    Re: Explanation of Solution Help

    Quote Originally Posted by petenice View Post
    My online class provided this solution but i have questions about some of the steps. My online professor has been no help. Any input is appreciated.

    Question Solve the initial value problem. \frac{dy}{dx}=\frac{y-1}{x+3}

    Step 1 Separate and integrate \int\frac{dy}{y-1}=\int\frac{1}{x+3}dx

    Step 2 Integration result ln(y-1)=ln(x+3)+lnC So where did the lnC come from. I know where the C came from but why are they multiplying it by ln

    Step 3 Raise all to the e e^{ln(y-1)}=e^{ln(x+3)}+e^{lnC}

    Step 4 Result is (y-1)=C(x+3) Why are they multiplying the right side by C shouldnt it be like this (y-1)=(x+3)+C

    Step 5 (y-1)=-C(x+3) why it the right side now negative?

    Step 6 y=1-C(x+3) Solution given to us by the professor.

    I cant figure out how they arrived to some of those steps maybe someone here can. Thanks.
    this is a mess. start from step 2 with a small change

    \ln(y-1)=\ln(x+3)+C_1$ where $C_1 \in \mathbb{R}

    now take the exponential on both sides

    y-1=e^{ln(x+3)+C_1}=e^{C_1}(x+3)

    y=1+e^{C_1}(x+3)

    now if you like you can choose C=\ln(\left|C_1\right|) and the resulting equation is

    y=1+C(x+3)

    note that C can be any number so it doesn't matter whether you use plus or minus in front of it so

    y=1-C(x+3) is equally valid.
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  3. #3
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    Re: Explanation of Solution Help

    Quote Originally Posted by petenice View Post
    Step 2 Integration result ln(y-1)=ln(x+3)+lnC So where did the lnC come from. I know where the C came from but why are they multiplying it by ln
    Hold on a minute. The integration result is NOT \displaystyle \begin{align*} \ln{ \left( y - 1 \right) } = \ln{ \left( x + 3 \right) } + \ln{ \left( C \right) } \end{align*}, it's \displaystyle \begin{align*} \ln{ \left| y - 1 \right| } = \ln{ \left| x + 3 \right| } + \ln{ \left( C \right) } \end{align*}.

    Also, we are NOT "multiplying by ln". Do you understand what "ln" means? It is taking the natural (base e) logarithm! Now the reason they are writing it as \displaystyle \begin{align*} \ln{ \left( C \right) } \end{align*} is simply to aid in calculations, and noting that since the integration constant is ARBITRARY, and \displaystyle \begin{align*} \ln{ \left( C \right) } \end{align*} is still a constant, it is just as correct to write a logarithm constant.

    Step 3 Raise all to the e e^{ln(y-1)}=e^{ln(x+3)}+e^{lnC}

    Step 4 Result is (y-1)=C(x+3) Why are they multiplying the right side by C shouldnt it be like this (y-1)=(x+3)+C
    NO! The rules of exponentials do not work like that! What should be happening is to collect the logarithms onto one side and to simplify using logarithm laws first BEFORE exponentiating!

    \displaystyle \begin{align*} \ln{ \left| y - 1 \right| } &= \ln{ \left| x + 3 \right| } + \ln{ \left( C \right) } \\ \ln{ \left| y - 1 \right| } - \ln{ \left| x + 3 \right| } &= \ln{ \left( C \right) } \\ \ln{ \left| \frac{y - 1}{x + 3} \right| } &= \ln{ \left( C \right) } \end{align*}

    Now exponentiate.

    The solutions given by your professor are VERY sloppy. Feel free to tell him/her that. In my opinion, you need to get the hang of trying a few steps out yourself without relying too much on following step-by-step solutions given by others. You know when a step is valid or not valid, and there's no reason to believe that when you have done all valid steps, that your answer, while it may look different to your professor's or others', is not correct.
    Thanks from LimpSpider and jguzman
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    Re: Explanation of Solution Help

    Agree with the sloppy Professor statement.
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    Re: Explanation of Solution Help

    Thanks guys. Prove It, i appreciate the breakdown. I just wanted to make sure I wasn't missing something. So far this professor has been less than helpful, but i appreciate everyones help.
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    Re: Explanation of Solution Help

    Quote Originally Posted by LimpSpider View Post
    Agree with the sloppy Professor statement.
    But we don't know how much of that was the Professor and how much was petenice's memory of it.
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