Variation - Variation of Parameters Recall that our last...

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Variation of Parameters Recall that our last theorem states that the nonhomogeneous equation y 00 + p ( x ) y 0 + q ( x ) y = g ( x ) has the general solution y ( x ) = ψ ( x ) + c 1 y 1 ( x ) + c 2 y 2 ( x ) where ψ ( x ) is a particular solution to the nonhomogeneous equation and c 1 y 1 ( x ) + c 2 y 2 ( x ) is the general solution to the nonhomogeneous equation. As a result, we would like a way to generate a particular solution ψ ( x ) , assuming we have already solved the somewhat easier homogeneous equation. That is, given a fundamental set of solutions y 1 and y 2 to the homogeneous equation, can we come up a particular solution ψ to the nonhomogeneous equation. The answer is yes, and the following method illustrates how this is possible. Variation of Parameters We will follow a strategy similar to the one we used in the Reduction of Order method, in the sense that we will try a solution of the form ψ ( x ) = u 1 ( x ) y 1 ( x ) + u 2 ( x ) y 2 ( x ) and see if we can find conditions on the functions u 1 and u 2 so that ψ is a particular solution to the nonhomogeneous equation.
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