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Math Help - differential equations

  1. #1
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    Exclamation differential equations

    find a general solution to the differential equation

    y*ln(x)-xy' =0

    this is my work so far:
    ylnx=x dy/dx
    ylnx dx= xdy
    (lnxdx)/x= dy/y
    integral of (lnx)/x dx = integral of dy/y

    ???? = lny

    i need help finding the integral of (lnx)/x dx
    and then i need to solve for y

    thank you!
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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

    y*ln(x)-xy' =0

    this is my work so far:
    ylnx=x dy/dx
    ylnx dx= xdy
    (lnxdx)/x= dy/y
    integral of (lnx)/x dx = integral of dy/y

    ???? = lny

    i need help finding the integral of (lnx)/x dx
    and then i need to solve for y

    thank you!
    Try  u = \ln x in \int \frac{\ln x}{x}\, dx
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  3. #3
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    oh thank you! i was on winter break for 2 weeks and i guess i'm fuzzy on integrating
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    wait so what do i do after i get u=lnx and du= 1/x dx
    is it 1/2 u^2 +c
    1/2 (lnx)^2 +c ??
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  5. #5
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    Quote Originally Posted by holly123 View Post
    wait so what do i do after i get u=lnx and du= 1/x dx
    is it 1/2 u^2 +c
    1/2 (lnx)^2 +c ??
    Yep - that's right!
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  6. #6
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    thanks!! so i have to inject e to get rid of ln and solve for y
    would that be y= 1/2 x^2 + e^c
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    Quote Originally Posted by holly123 View Post
    thanks!! so i have to inject e to get rid of ln and solve for y
    would that be y= 1/2 x^2 + e^c
    You can't simplify like that. In general

    \left( \ln x \right)^n \ne \ln \left( x^n \right)

    and

     a \ln (x) \ne \ln (a x)
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  8. #8
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    for some reason this is really confusing me...i tried again and got y=x+ e^c
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  9. #9
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    Quote Originally Posted by holly123 View Post
    for some reason this is really confusing me...i tried again and got y=x+ e^c
    Since  \ln y = \frac{1}{2} \left( \ln x \right)^2 + c

    then the best you're going to get is

    y = e^{\frac{1}{2} \left( \ln x \right)^2} \cdot e^c = C e^{\frac{1}{2} \left( \ln x \right)^2} where C = e^c
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