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# Variation of parameters (solving inhomo. linear ODEs) watch

1. 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?
2. (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

In comes a load of LaTeX! Bear in mind that must satisfy

Additionally, set and to keep things concise.

The claim is that is a particular solution to the ODE. All you need to do to check this is differentiate:

Plug this into the initial equation and we get:

The important thing to notice here, is that since satisfy the corresponding homogenous equation, we have for any

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

Hence we are left with

Then we go on to solve for . Once this is done, is satisfied and hence the original equation is satisfied, so is indeed a particular solution to the ODE.
3. 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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