(d^2 +2d +1)y=e^-x log(x) solve using method variation of parameter?

First you'll need the complementary solutions. The characteristic equation for this ODE is



with multiplicity 2. This means two linearly independent solutions will be and .

Via variation of parameters, the particular solution will take the form



where




The Wronskian is



So, you have





and





So the solution to the ODE is

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