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Math Help - homogenous equation

  1. #1
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    homogenous equation

    please give me a hint how to start this@

    x(dy/dx) = y + 2xsin^2(y/x)

    basically i dont know what the first step is because the way i learnt this, i can't see how to apply that method to this example?
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  2. #2
    A Plied Mathematician
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    You have two equals signs there. Did you mean a plus sign in there somewhere?
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  3. #3
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    yes sorry it is i edited it now
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  4. #4
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    ok i know to divide through x, sub z=y/x. but then how do you find dy/dx? if y=xz? is it chain rule? can someone talk me through this?
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  5. #5
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    i have solved it down to this

    -1/(2tan(y/x)) = lnx + c

    how can i isolate y please?
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  6. #6
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    isolate y

    i have an implicit solution that i need to make plicit

    -1/(2tan(y/x)) = lnx + c

    how can i isolate y please?
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  7. #7
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    \displaystyle \frac{-1}{2\tan\frac{y}{x}} = \ln x + c

    \displaystyle \frac{-1}{\ln x+ c} = 2\tan\frac{y}{x}

    \displaystyle \frac{-1}{2\ln x+ c} = \tan\frac{y}{x}

    \displaystyle \tan^{-1}\left(\frac{-1}{2\ln x+ c}\right) = \frac{y}{x}

    \displaystyle x\tan^{-1}\left(\frac{-1}{2\ln x+ c}\right) = y
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  8. #8
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    thanks, a lot

    should the constant be written as lnc so ln(x + c)?
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  9. #9
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    Quote Originally Posted by mathcore View Post
    thanks, a lot

    should the constant be written as lnc so ln(x + c)?
    \ln{x} + \ln{c} \ne \ln(x+c)

    \ln{x} + \ln{c} = \ln(cx)
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  10. #10
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    too right bro. it is, i forgot my log rules but still, i ask the question, is it better to leave the constant as c or stick it in with the ln (x)? i think putting it as ln(cx) seems neater than ln(x) + c? what is convention?
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  11. #11
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    Either way is fine, in general I notice \displaystyle \ln x + c = \ln x + \ln c = \ln (cx) when solving D.E.s

    It really depends on the situation and what is required.
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  12. #12
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    i think it is neater
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  13. #13
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    Different people learn this method different ways. I learned it this way:

    \displaystyle x\,\frac{dy}{dx} = y + 2x\sin^{2}\!\left(\dfrac{y}{x}\right)

    \displaystyle x\,dy-\left(y+2x\sin^{2}\!\left(\dfrac{y}{x}\right)\righ  t)dx=0.

    If we follow your substitution of z=y/x, then y=zx, and dy=z\,dx+x\,dz. (This is how I "find" dy/dx. If you wish, you can write dy/dx=z+x(dz/dx).) Plugging into the DE yields

    x(z\,dx+x\,dz)-(zx+2x\sin^{2}(z))\,dx=0

    x^{2}\,dz-2x\sin^{2}(z)\,dx=0

    x\,dz-2\sin^{2}(z)\,dx=0. Separating out yields

    \csc^{2}(z)\,dz-\dfrac{2\,dx}{x}=0

    -\cot(z)-2\ln|x|=C

    \cot(z)+2\ln|x|=C (absorbing the minus sign into the constant).

    \cot\left(\dfrac{y}{x}\right)=-2\ln|x|+C.

    At this point, as with all solutions to any differential equation, you should verify that this implicitly defined function satisfies the original DE. The reason to differentiate now is that if you do, the constant will go away, which is what you want when you're verifying the solution to a DE. The solution I've got here is, incidentally, the same as yours, I believe. So far, so good. If you do want to isolate y, then take the \cot^{-1} of both sides, and then multiply the resulting equation by x to obtain

    y=x\cot^{-1}\left(-2\ln|x|+C\right).

    On what interval(s) is this solution valid?
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