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Math Help - non-linear 2nd order differential

  1. #1
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    Unhappy non-linear 2nd order differential

    y'' + 2x (y')^2= 0

    show the general solution, then find an explicit formula for a solution y(x), which also satisfies intial conditions y'(0) =1 and y (0) = 0

    where to start?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by harveyo View Post
    y'' + 2x (y')^2= 0

    show the general solution, then find an explicit formula for a solution y(x), which also satisfies intial conditions y'(0) =1 and y (0) = 0

    where to start?
    Let u=y^{\prime}\implies u^{\prime}=y^{\prime\prime}.

    The equation becomes u^{\prime}+2xu^2=0\implies u^{\prime}=-2xu^2. Apply separation of variables to solve for u, where y^{\prime}\!\left(0\right)=1\implies u\!\left(0\right)=1.

    Then back substitute and solve the resulting differential equation for y.

    Can you take it from here?
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  3. #3
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    seperation of variables =

    u= -2x u^2

    dy/dx = -2x u^2

    dy/u^2 = -2x dx

    integrate both sides

    ln u^2 = -x^2 + C

    am i doing this correct
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by harveyo View Post
    seperation of variables =

    u= -2x u^2

    dy/dx = -2x u^2

    dy/u^2 = -2x dx

    integrate both sides

    ln u^2 = -x^2 + C

    am i doing this correct
    no. \int \frac 1{u^2}~du ~{\color{red} \ne }~ \ln u^2 + C (!)

    Note that \int \frac 1{u^2}~du = \int u^{-2}~du

    by the way, why do you have dy? it should be du
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  5. #5
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    du/dx = -2x u^2

    du/u = -2x dx

    intergrate u ^-2 du = intergrate -2x dx

    -u ^-1 = -x^+c
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  6. #6
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    Quote Originally Posted by harveyo View Post
    du/dx = -2x u^2

    du/u = -2x dx
    You mean du/u^2

    intergrate u ^-2 du = intergrate -2x dx

    -u ^-1 = -x^+c
    Good. so u= \frac{1}{x- c}

    And now, since u= \frac{dy}{dx}, you still need to solve
    \frac{dy}{dx}= \frac{1}{x- c}
    That should be an easy integral and will introduce a second "constant of integration".
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