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Math Help - implicit form

  1. #1
    Junior Member
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    implicit form

    Hi everyone,

    If anyone could help to check if no mistake in my calculation.

    Here is the question and my answer:

    Hence find, an implicit form, the general solution of the differential equation

    dy/dx = 2y3/2 cos x
    (2 + sin x)2
    The differential equation dy/dx = 2y3/2 cos x/(2+sin x)2,


    Has a right-hand side of the form f(x) g(y), where

    g(y) = 2y 3/2 cos x and f(x) = (2+sin x)2

    Dividing both sides of the differential equation by g(y) means multiplying both sides by y. This give

    ∫ 2y 3/2 cos x dy = ∫(2+sin x)2 dx
    y5/2-sin x + a = cos x3 + b

    Thus the general solution, in implicit form, is

    y 5/2 = cos x3 + c



    Thanks for your help
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by valerie-petit View Post
    Hi everyone,

    If anyone could help to check if no mistake in my calculation.

    Here is the question and my answer:

    Hence find, an implicit form, the general solution of the differential equation

    dy/dx = 2y3/2 cos x
    (2 + sin x)2
    The differential equation dy/dx = 2y3/2 cos x/(2+sin x)2,


    Has a right-hand side of the form f(x) g(y), where

    g(y) = 2y 3/2 cos x and f(x) = (2+sin x)2

    Dividing both sides of the differential equation by g(y) means multiplying both sides by y. This give

    ∫ 2y 3/2 cos x dy = ∫(2+sin x)2 dx
    y5/2-sin x + a = cos x3 + b

    Thus the general solution, in implicit form, is

    y 5/2 = cos x3 + c



    Thanks for your help
    Thse are just seperable differential equations

    All you need to do is this

    it will be of the form

    \frac{dy}{dx}=f(x)g(y)

    So you need to seperate to make it look like

    g(y)dy=f(x)dx

    Which implies that

    \int{g(y)}dy=\int{f(x)dx}

    Now let G(y) and F(x) be anti-derivatives respectively.

    Then we have that

    G(y)=F(x)+C

    This implies that

    y=G^{-1}\left(F(x)+C\right)

    Where G^{-1} denotes the inverse function of G(y)
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  3. #3
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    Lexington, MA (USA)
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    Hello, Valerie!

    If I read your typing correctly, you went way off . . .


    Find, in implicit form, the general solution of the differential equation:

    . . \frac{dy}{dx} \;=\; \frac{2y^{\frac{3}{2}}\cos x}{(2 + \sin x)^2}
    Separate variables ... divide by y^{\frac{3}{2}}

    . . y^{-\frac{3}{2}} \;=\;\frac{2\cos x\,dx}{(2 + \sin x)^2}

    Integrate: . -2y^{-\frac{1}{2}} \;=\;-2(2+\sin x)^{-1} + C \quad\Rightarrow\quad y^{-\frac{1}{2}} \;=\;(2+\sin x)^{-1} + C

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  4. #4
    Junior Member
    Joined
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    Re: implicit form

    Thanks Soroban for your help.
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  5. #5
    Junior Member
    Joined
    Jun 2008
    Posts
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    Explicit form

    On this differential equation I was asked to find a particular solution with y= 4 and x= 0. Which I did, now I have to find the explicit form of that particular solution.

    Here is the question and my answer:

    Find the explicit form of this particular solution.
    y-1/2 = (2+sin x)-1 + 3/8



    The explicit solution is obtained by making y the subject of the equation




    y= (2 + sin x)^1/2 +3/8



    I multiplied -1/2 both side
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