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Math Help - Diff. eqn + erf (error function)

  1. #1
    veronik
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    Question Diff. eqn + erf (error function)

    Iím stacked with this problem for many days, someone can help me pleeeeease:

    (a) f \left( x \right) =\int _{-\infty }^{{x}^{2}/2}\!{e^{x-1/2\,{t}^{2<br />
}}}{dt}

    I foud the solution: f \left( x \right) =1/2\,{e^{x}}\sqrt {2\pi } \left( 1+{\it <br />
erf} \left( 1/4\,{x}^{2}\sqrt {2} \right)  \right)

    (b) Find the solution of the dfferential equatio:
    {\frac {d^{2}}{d{x}^{2}}}y \left( x \right) =f \left( x \right) with y(0)=0 and dy(0)/dx = 0

    In the form : y \left( x \right) =\int _{0}^{x}\! \left( x-t \right) f \left( t<br />
 \right) {dt}

    Veronica
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  2. #2
    Grand Panjandrum
    Joined
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    Quote Originally Posted by veronik View Post
    I’m stacked with this problem for many days, someone can help me pleeeeease:

    (a) f \left( x \right) =\int _{-\infty }^{{x}^{2}/2}\!{e^{x-1/2\,{t}^{2<br />
}}}{dt}

    I foud the solution: f \left( x \right) =1/2\,{e^{x}}\sqrt {2\pi } \left( 1+{\it <br />
erf} \left( 1/4\,{x}^{2}\sqrt {2} \right)  \right)

    (b) Find the solution of the dfferential equatio:
    {\frac {d^{2}}{d{x}^{2}}}y \left( x \right) =f \left( x \right) with y(0)=0 and dy(0)/dx = 0

    In the form : y \left( x \right) =\int _{0}^{x}\! \left( x-t \right) f \left( t<br />
 \right) {dt}

    Veronica
    There is nothing to do other than show that:

    y( x ) =\int _{0}^{x}\! ( x-t ) f( t) {dt}

    satisfies the differential equation, as this already satisfies the initial conditions.

    You do this by differentiating the proposed solution twice, and so obtain the
    given differential equation.

    You will need to use the following a number of times:

    \frac{d}{dx} \int_0^x g(t) ~dt=g(x)

    RonL
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