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Math Help - Differential equations

  1. #1
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    Differential equations

    I'm after some help with these two equations.

    My problem is factorising with equations like this, I can never tell when I'm supposed to do it!! I have the answers to these questions I just need explicit help, basically a walk through of these examples, if anyone can help I'd much appreciate it,
    Thanks

    ydy/dx+7x^2+7x+5=0

    which satisfies y(0)=4

    The Answer is:
    sqrt(16-(14/3x^3 + 7x^2+10x))



    and my other problem is,



    dy/dx = 1+y^2/7+9x

    which satisfies y(1) = 0

    tan(1/9ln(|7+9x/16|))



    With regards to the last answer, how does PI sometimes come out of this answer?


    PS how do you write with math symbols, it looks a bit messy....
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    Both of them are separable.

    (See my signature for LaTeX typesetting.)
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  3. #3
    Super Member wingless's Avatar
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    A separable differential equation is an equation that can be expressed in this form:
    \frac{dy}{dx} = g(x)\cdot h(y)

    Like, \frac{dy}{dx} = 4xy^2,~ \frac{dy}{dx} = \frac{x^2y-y}{x+1},~ \frac{dy}{dx} = \frac{xy^3}{\sqrt{x^2+1}}~.~.~.

    You can solve them by separating the functions and differentials to the sides and integrating.

    For example,

    \frac{dy}{dx} = 2xy^2

    Move dx to the right and y^2 to the left.

    \frac{1}{y^2}~dy = 2x~dx

    Now you can integrate both sides.

    \int \frac{1}{y^2}~dy = \int 2x~dx

    -\frac{1}{y} + C_1 = x^2 + C_2

    We can use only C instead of two integration constants.
    -\frac{1}{y} = x^2 + C

    y = - \frac{1}{x^2 + C}

    -------------------------------------

    Your first question is,
    y\frac{dy}{dx} +7x^2+7x+5=0, y(0)=4

    Start by separating.

    y~dy = -7x^2-7x-5~dx

    Now we can integrate both sides.

    \int y~dy = \int -7x^2-7x-5~dx

    \frac{y^2}{2}+C_1 = -\frac{7}{3}x^3 -\frac{7}{2}x^2-5x +C_2

    \frac{y^2}{2} = -\frac{7}{3}x^3 -\frac{7}{2}x^2-5x +C

    y^2 = -\frac{14}{3}x^3 -7x^2-10x +2C

    |y| = \sqrt{-\frac{14}{3}x^3 -7x^2-10x +2C}

    ----- y = \sqrt{-\frac{14}{3}x^3 -7x^2-10x +2C}

    ----- y = -\sqrt{-\frac{14}{3}x^3 -7x^2-10x +2C}

    Try y(0) = 4 for both of them.

    ----- y = \sqrt{-\frac{14}{3}x^3 -7x^2-10x +2C}
    ----- 4 = \sqrt{2C}
    ----- \boxed{2C = 16}

    ----- y = -\sqrt{-\frac{14}{3}x^3 -7x^2-10x +2C}
    ----- 4 = -\sqrt{2C}
    -----As \sqrt{2C} is always positive, this equation doesn't have a root.

    So we found that our function is y = \sqrt{-\frac{14}{3}x^3 -7x^2-10x +2C} because this is the only valid function that makes y(0) = 4. And we found that 2C = 16.

    Hence, the function we are looking for is:
    y = \sqrt{-\frac{14}{3}x^3 -7x^2-10x + 16}

    y = \sqrt{16-(\frac{14}{3}x^3 +7x^2+10x)}


    -------------------------------------

    Now try to solve the second equation
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