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Variation of parameters (solving inhomo. linear ODEs) Watch

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    Hi, there is something I don't quite understand in the method of variation of parameters to solve inhomogeneous linear ODEs. I will just refer to the Wikipedia page:

    http://en.wikipedia.org/wiki/Variation_of_parameters

    What I don't understand is: if y1(x),...yn(x) form a fundamental system for the associated linear ODE, why does that imply the general solution of the original nonhomo. linear ODE is

    c1(x)y1(x)+...+cn(x)yn(x)

    where c1(x),...,cn(x) are continuous functions?
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    (Original post by Kelvinator)
    What I don't understand is: if y1(x),...yn(x) form a fundamental system for the associated linear ODE, why does that imply the general solution of the original nonhomo. linear ODE is

    c1(x)y1(x)+...+cn(x)yn(x)

    where c1(x),...,cn(x) are continuous functions?
    Don't forget that those continuous functions must satisfy several conditions, they are not arbitrary functions. Once they do, it just comes down to a bit of differentiation. Also, that sum is not the general solution, but a particular solution.

    The general solution is achieved by adding the particular solution to \sum y_i

    In comes a load of LaTeX! Bear in mind that c_1,c_2\cdots c_n must satisfy \displaystyle\sum_{i=1}^n c_i'(x)y_i^{(j)}(x)=0\quad(\star  )

    Additionally, set c_0(x)=y_0(x)=0 and a_n(x)=1 to keep things concise.

    The claim is that y_p(x)=\displaystyle\sum_{i=0}^n c_i(x)y_i(x) is a particular solution to the ODE. All you need to do to check this is differentiate:

    y_p^{(1)}(x)=\displaystyle \underbrace{\sum_{i=0}^n c_i'(x)y_i(x)}_{=\;0\; \Leftarrow\, (\star)}+\sum_{i=0}^n c_i(x)y_i^{(1)}(x)


    \cdots

    y_p^{(n-1)}(x)=\displaystyle \underbrace{\sum_{i=0}^n c_i'(x)y_i^{(n-2)}(x)}_{=\;0\; \Leftarrow\, (\star)}+\sum_{i=0}^n c_i(x)y_i^{(n-1)}(x)
    y_p^{(n)}(x)=\displaystyle \sum_{i=0}^n c_i'(x)y_i^{(n-1)}(x)+\sum_{i=0}^n c_i(x)y_i^{(n)}(x)

    Plug this into the initial equation and we get:

    \displaystyle\sum_{i=0}^n c_i'(x)y_i^{(n-1)}(x)+\sum_{i=0}^{n}a_i(x)\sum_  {k=0}^i c_k(x) y_k^{(i)}(x)=b(x)\quad (\star\star)

    The important thing to notice here, is that since y_0,y_1\cdots y_n satisfy the corresponding homogenous equation, we have for any 0\leq k\leq n:

    \displaystyle\sum_{i=0}^{n}a_i(x  )y_k^{(i)}(x)=0

    In other words, the latter sum in (\star\star) is  0 (since the c_k(x) are continuous, they can have no effect on this)

    Hence we are left with \displaystyle\sum_{i=0}^n c_i'(x)y_i^{(n-1)}=b(x)\quad (\clubsuit)

    Then we go on to solve for c_i'. Once this is done, (\clubsuit) is satisfied and hence the original equation is satisfied, so y_p is indeed a particular solution to the ODE.
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    Thanks for the explanation. I was confused because I had it in my mind that your y_p is the general solution but turns out I misunderstood my lecturer when he said "we will find the solution of the form...". But now it all makes sense.
 
 
 
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