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Thread: sorry, another differential equation

  1. #1
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    Unhappy sorry, another differential equation

    find a general solution to the differential equation

    (2+x) dy/dx = 3y

    i got the answer y= 6+3x+e^(3c)
    but i don't think that makes much sense..
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  2. #2
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    Quote Originally Posted by holly123 View Post
    find a general solution to the differential equation

    (2+x) dy/dx = 3y

    i got the answer y= 6+3x+e^(3c)
    but i don't think that makes much sense..
    The differential equation separates
    i.e.

    $\displaystyle \frac{dy}{y} = 3 \frac{dx}{x+2}$

    so

    $\displaystyle \ln |y| = 3 \ln |x +2| + c$

    so how did you get your answer (careful)?
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  3. #3
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    okay thats what i got but i didn't know how to solve for y. i know you have to inject e to cancel out ln
    so would it be y= 3(2+x) + e^(3c)
    im not sure
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  4. #4
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    Quote Originally Posted by danny arrigo View Post
    The differential equation separates
    i.e.

    $\displaystyle \frac{dy}{y} = 3 \frac{dx}{x+2}$

    so

    $\displaystyle \ln |y| = 3 \ln |x +2| + c$

    so how did you get your answer (careful)?
    So $\displaystyle e^{\ln |y|} = e^{3 \ln |x +2| + c}$ so

    $\displaystyle e^{\ln |y|} = e^{\ln |x +2|^3 + c} = e^{\ln |x +2|^3} \cdot e^{c} $

    giving

    $\displaystyle y = k (x+2)^3$ where $\displaystyle k = e^c$
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  5. #5
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    Hello, holly123!

    Find a general solution to the differential equation: .$\displaystyle (2+x)\frac{dy}{dx} \:= \:3y$

    i got the answer: .$\displaystyle y\:=\: 6+3x+e^{3c}$
    but i don't think that makes much sense. . . . . no, it doesn't

    Separate the variables: .$\displaystyle \frac{dy}{y} \:=\:\frac{3\,dx}{x+2}$

    Integrate: .$\displaystyle \int\frac{dy}{y} \:=\:3\int\frac{dx}{x+2} \quad\Rightarrow\quad \ln y \:=\:3\ln(x+2) + c$

    . . $\displaystyle \ln y \:=\:\ln(x+2)^3 + \ln C \quad\Rightarrow\quad \ln y \:=\:\ln\bigg[C(x+2)^3\bigg] $


    Therefore: .$\displaystyle y \:=\:C(x+2)^3$

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  6. #6
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    thank you
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