On the basis of your hypotheses is...
... so that setting , because the derivative is a linear operator, you have...
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let y_p be a particular solution of the nonhomogeneous eqn y'' + py' + qy = f(x) and let y_c be a solution of its associated homogeneous eqn. show that y=y_c + y_p is a solution of the given nonhomogeneous eqn.