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Math Help - General Solution of Differenatial Equation

  1. #1
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    General Solution of Differenatial Equation

    I need help with finding the general solution of this differenatial equation:

    1/x^6 dy/dx = 19 + 9y

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  2. #2
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    Re: General Solution of Differenatial Equation

    Hello, MMCS!

    Separate the variables . . . and integrate.
    Thanks from Ruun
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  3. #3
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    Re: General Solution of Differenatial Equation

    Multiply by x^6

    \frac{dy}{dx}=x^6(19+9y)

    Divide by 19+9y

    \frac{dy}{dx(19+9y)}=x^6

    Integrate

    \int\frac{dy}{19+9y}=\int x^6 dx

    \int \frac{dy}{19+9y}=\int\frac{dy}{9(\frac{19}{9}+y)}=  \frac{1}{9}\ln(\frac{19}{9}+y)+C

    \int x^6dx=\frac{x^7}{7}+\bar{C}

    \frac{1}{9}\ln(\frac{19}{9}+y)+C=\frac{x^7}{7} + \bar{C} \rightarrow \ln(\frac{19}{9}+y)=\frac{9x^7}{7}+D

    Taking exponentials

    \frac{19}{9}+y=e^{\frac{9x^7}{7}+D}=Ke^{\frac{9x^7  }{7}}


    So

    \boxed{y=Ke^{\frac{9x^7}{7}}-\frac{19}{9}, K \in \mathbb{R}}

    Checking the answer:

    \frac{dy}{dx}=Ke^{\frac{9x^7}{7}}9x^6


    Noting that

    Ke^{\frac{9x^7}{7}}=y+\frac{19}{9}

    \frac{dy}{dx}=\left(y+\frac{19}{9}\right)9x^6=(9y+  19)x^6

    As requested
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    Re: General Solution of Differenatial Equation

    That's the perfect answer.
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  5. #5
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    Re: General Solution of Differenatial Equation

    perfect Ruun, Thanks!
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