Is the first condition in the method of variation of parameters really free?

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James Leigh

suppose v1'y1 + v2'y2 = constant (eqn A)

then the second linear equation between v1' and v2' obtained by substituting in the original DE is unaltered, because this only depends on the differentiation of A and this is no different to the case where constant is zero. ie eqn is v1'y1' + v2'y2' = R(x)

on integrating you just get a different particular integral.