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Why is variation of parameters ever useful relative to reduction of order?

Reduction of order can solve any linear ODE given a single, particular solution to the associated homogeneous ODE. Variation of parameters can solve any linear ODE given the general solution to the associated homogeneous ODE. It should always be at least as difficult, and usually more difficult, to find a general homogeneous solution than to find a particular homogeneous solution. This sounds like a higher barrier to entry with no extra reward.

Why use variation of parameters to solve second or higher order linear ODEs at all? Can it solve any ODEs that reduction of order can't?

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Reduction of order can only reduce the order by 1 for each known solution. It doesn't automatically solve anything, you need to make enough guesses for it to work.

Really, both methods are the same, there are no essential differences between them, they are just dressed up differently. This is easier to see when you convert an order n equation into an equation in dimension n. Then solving an inhomogeneous equation is equivalent to solving a homogeneous equation in 1 dimension higher where 1 solution is already know (using adjoint matrix). Then the determinant of the Wronskian can be found by solving the first order homogeneous equation associated to the Wronskian. Finally, if all except one column are known and the determinant is known, then a valid possible last column can be found using Cramer's rule. And this is how both method works.

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