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Math Help - Help me to solve a second order differential equation.

  1. #1
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    Smile Help me to solve a second order differential equation.

    Can any one help me to solve a differential equation of the form

    d2y/dx2 + f(x) y = 0

    where, f(x) can be any kind of function
    e.g. f(x)=sin2x or exp(x) or ax2+bx+c or cosech(x) etc.
    Attached Thumbnails Attached Thumbnails Help me to solve a second order differential equation.-diff-eqn.jpg  
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  2. #2
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    Quote Originally Posted by skullface View Post
    Can any one help me to solve a differential equation of the form

    d2y/dx2 + f(x) y = 0

    where, f(x) can be any kind of function
    e.g. f(x)=sin2x or exp(x) or ax2+bx+c or cosech(x) etc.
    ODE's without constant coefficients are hard to find general solutions to.
    One common method is to attempt to find a power series solution centered around your initial conditions. (The other is numerically)

    It usually goes something like this suppose that

    Note that I am using the y_n to be the constants in the series expansion

    \displaystyle y(x)=\sum_{n=0}^{\infty}y_nx^n \implies y(x)''=\sum_{n=0}^{\infty}(n+1)(n+2)y_{n+2}x^n

    Now we need to do the same thing with f(x) this gives

    \displaystyle f(x)=\sum_{n=0}^{\infty}f_nx^n

    Now comes the bad part we need to product of the series
    f(x)y(x)

    \begin{array}{c  c c c c c c c c c c c} <br />
\,& (f_0 &+& f_1x &+&f_2x^2&+& \cdots & +&f_nx^n&+&\cdots )\\<br />
\times &(y_0 &+& y_1x &+&y_2x^2&+& \cdots & +&y_nx^n&+&\cdots )\\<br />
\hline \,&y_0f_0 &+& y_0f_1x &+&y_0f_2x^2&+& \cdots & + & y_0f_nx^n & + & \cdots  \\<br />
 \,& \, &\,& y_1f_0x &+&y_1f_1x^2&+& \cdots & +&y_1f_{n-1}x^n&+&\cdots  \\<br />
 \,& \, &\,& \, &\,&y_2f_0x^2&+& \cdots & +&y_2f_{n-2}x^n&+&\cdots  \\<br />
\end{array}

    Using the columns gives the series representation

    \displaystyle f(x)y(x)=\sum_{n=0}^{\infty}\left(\sum_{i=0}^{n} y_i \cdot f_{n-i}\right)x^n

    Now just equate coefficients

    \displaystyle y''(x)=-f(x)y(x)

    This gives

    \displaystyle (n+1)(n+2)y_{n+2}=\left(\sum_{i=0}^{n} y_i \cdot f_{n-i}\right) \iff y_{n+2}=\left(\sum_{i=0}^{n} \frac{y_i \cdot f_{n-i}}{(n+1)(n+2)}\right)

    Now you should know all of the f_n and the first two y_n your initial conditions and this will give a recurrence relation for all of the coefficients of the power series.
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  3. #3
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    u might also "guess" the solution is of kind:

    <br />
y=e^{g(x)}<br />
    and get that:
    <br />
g` + g^2 + f = 0<br />

    and then u'll need to solve this, but the solution method depends on f
    (or u can apply the "duhamel principle", and then the solution is global.
    i don't remember rules for applying it)
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